Yield Curve Modeling under Cyclical Influence
Mats Levander
January 2010
Abstract
There are few models for long-term yield forecasting and especially in the
Real World Measure. This thesis uses Principal Component Analysis (PCA)
to analyze the yield curves and gives an update of precedent studies. The
conclusion is still that the first three components is enough to describe the
variation of the yield curve. For simulation of the yield curves PCA and a
semi parametric approach are evaluated. These models fail to yield plausible simulations. Therefore a new model is developed. The model is influenced by a business cycle and a relationship is derived from historical data
between the yield curve and the cycle. Nelson-Siegel’s model and OrnsteinUhlenbeck processes are some of the features used in the model. Tools are
also introduced to cope with the non-negativity problems in the aftermath
of the financial crisis. The model will be used as input to calculate the Swap
Breakage Exposure upon an early termination of an interest rate swap.
Acknowledgements
I would like to thank Nils Hallerström at PK AirFinance for giving me
this opportunity and supporting me by sharing his wide knowledge of the
financial markets. I would also like to thank the whole PK AirFinance Office in Luxembourg for baring with me during this period, and especially
Christophe Beaubron, Frank Withofs and Bo Persson for guidance and discussions. Finally, I would like to thank my supervisor at KTH, Henrik Hult,
for his support and ideas during the course of this thesis.
Stockholm, January 2010
Mats Levander
ii
Contents
1 Introduction
1.1 PK AirFinance . . . . . . . . . . . .
1.2 SAFE . . . . . . . . . . . . . . . . .
1.3 The Aircraft Value Cycle . . . . . .
1.3.1 Introduction . . . . . . . . .
1.3.2 Historical evidence . . . . . .
1.3.3 Why this behavior? . . . . .
1.3.4 Amplitude of Cyclical Swings
1.3.5 Prediction of the cycle . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
2
7
7
7
8
9
9
2 Literature Study
13
3 Theoretical Background
3.1 Inflation . . . . . . . . . . . . . . . .
3.2 Present Value . . . . . . . . . . . . .
3.2.1 Compounding of rates . . . .
3.3 Forward Rates . . . . . . . . . . . .
3.4 Yield . . . . . . . . . . . . . . . . . .
3.4.1 Yield Curve . . . . . . . . . .
3.5 Interest Rate Swap . . . . . . . . . .
3.6 Swap Breakage . . . . . . . . . . . .
3.6.1 Example of Swap Breakage .
3.7 LIBOR rate . . . . . . . . . . . . . .
3.7.1 Historical Background . . . .
3.7.2 BBA-LIBOR . . . . . . . . .
3.7.3 Definition . . . . . . . . . . .
3.7.4 Applications . . . . . . . . .
3.7.5 Selection of panels . . . . . .
3.7.6 Calculation . . . . . . . . . .
3.8 U.S. Interest Rate Swaps . . . . . . .
3.9 US Government Debt . . . . . . . .
3.9.1 Treasury Bills . . . . . . . . .
3.9.2 Treasury Bonds and Treasury
17
17
18
18
19
19
20
20
21
21
23
23
24
24
25
25
26
26
26
26
27
iii
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
Notes
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.10 Smoothing Spline . . . . . . . . . . .
3.11 Correlation . . . . . . . . . . . . . .
3.11.1 Cross-Correlation . . . . . . .
3.11.2 Cross-Correlation Coefficient
3.12 Least Squares Method . . . . . . . .
3.13 Maximum Likelihood Method . . . .
3.14 Linear Regression . . . . . . . . . . .
3.14.1 The Simple Linear Model . .
3.14.2 Multivariate Models . . . . .
3.14.3 Goodness of Fit . . . . . . . .
3.14.3.1 R2 and adjusted R2
3.15 Principal Component Analysis . . .
3.16 Ornstein-Uhlenbeck process . . . . .
3.17 Nelson-Siegel’s Yield Curve Model .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
28
28
29
30
30
31
31
32
33
33
33
34
35
36
4 Data Selection
39
5 Results
5.1 Principal Component Analysis . . . . . . . . . . .
5.1.1 LIBOR and Swaps . . . . . . . . . . . . . .
5.1.1.1 Different Valuation Basis . . . . .
5.1.1.2 Weekly data . . . . . . . . . . . .
5.1.2 US Treasuries Constant Maturity . . . . . .
5.1.2.1 Daily Data . . . . . . . . . . . . .
5.1.2.2 Weekly data . . . . . . . . . . . .
5.1.2.3 Monthly Data . . . . . . . . . . .
5.1.2.4 Conclusion PCA . . . . . . . . . .
5.1.3 Correlation with the PURCS Cycle . . . . .
5.1.3.1 Crosscorrelation with PURCS . .
5.2 Simulation: PCA . . . . . . . . . . . . . . . . . . .
5.2.1 LIBOR and US Swaps . . . . . . . . . . . .
5.2.1.1 Nominal . . . . . . . . . . . . . .
5.2.1.2 Real . . . . . . . . . . . . . . . . .
5.2.2 US Treasuries Constant Time to Maturity .
5.3 Simulation: Rebonato . . . . . . . . . . . . . . . .
5.3.1 T-notes . . . . . . . . . . . . . . . . . . . .
5.4 Simulation: Brute Force Model . . . . . . . . . . .
5.4.1 Calibration of Ornstein-Uhlenbeck processes
41
41
41
44
46
47
47
51
52
53
53
56
56
57
57
62
63
65
65
68
74
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6 Discussion
81
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A PCA Results
85
iv
B Maximum Likelihood Estimation Ornstein-Uhlenbeck
95
vi
Chapter 1
Introduction
1.1
PK AirFinance
PK AirFinance started in 1979 when the Swedish bank PKbanken set up
a subsidiary, PK Finans, in Luxemburg. In 1983 PK Finans made their
first aircraft financing - a DC-10-30 to Martinair. A couple of years later
a separate Air Group was established. Crédit Lyonnais bought 85% of the
Air Group in 1988 and it was renamed to Crédit Lyonnais/PK AirFinance.
Under the following years the company expanded, e.g. it set up offices in
New York and Tokyo, started an Asset Management Group and a first version of a deal analysis model was launched. In 1991 Crédit Lyonnais bought
the remaining 15% of Air group.
In 2000 the company was bought by GECAS (General Electric Commercial Aviation Services) and renamed to PK AirFinance. An office was set
up in Toulouse in 2001 because of the proximity with Airbus. As of today PK AirFinance is a leading provider and arranger of asset based loans
that are secured by commercial jet aircraft. The company’s customers are
airlines, aircraft traders, lessors, investors, financial institutions and manufacturers worldwide.
PK AirFinance’s business activity is to; invest for its own account in aircraft
backed loans and securities. This is done by providing or purchasing; senior1
loans or finance leases, junior loans, and minor equity investments in special
purpose companies that own aircraft and guarantees.
The parent company, GECAS, is the world’s largest aircraft lessor. PK
AirFinance has tailor-made financing deals involving more than 75 Airlines
worldwide. At the end of 2008 the company’s portfolio volume reached
USD 7.7 billion. As of today PK AirFinance has 15 employees. The deal
1
a senior loan has higher priority than a junior loan in a repayment structure
1
analysis model, SAFE2 , was created in 1993. This tool has been subject to
continuous refinement during the years.
1.2
SAFE
PK AirFinance (PK Air) uses SAFE to analyze, price, underwrite and to
continuously monitor deals throughout their term in the portfolio. The
model is made up by several modules that contribute to the output of the
model. The model has a wide range of applicability, e.g. for determining
risk effectiveness, value creation and portfolio management. The graphical
user interface is in an Excel environment supported by C++ and databases.
In this interface PK Air can alter many different parameters in the analysis
of new or existing deals. Examples of parameters are jurisdiction, margins,
type of lease, aircraft type and many others. Some of these parameters are
analyzed in different subroutines. For example they model the future aircraft value, the probability of default of the obligor and the industrial cycle
in subroutines.
To put it short, the model boils down to calculation of the distribution
of the Net Present Value (NPV) of a deal. In this calculation PK Air generates many different scenarios of the outcome of the NPV. The risk in the
deals is when the NPV turns negative and PK Air will experience losses.
To value different deals PK Air uses different metrics such as Risk Reward
Ratio (Expected Loss divided by Expected Present Value), Average Downside Risk (the average of the negative outcomes in the NPV scenarios) and
Value At Risk (VaR).
The fundamentals of PK Air’s Asset Backed Financing are
• Default Risk
• Asset Value
• Deal Structure
Default Risk is the risk that the airline will default (not being able to meet
its payment obligations). The probability of default is linked to the credit
ratings of the airline. Another thing that affects the probability of default
is the Aircraft Value Cycle which is explained thoroughly in Section 1.3,
briefly it can be said to adjust for the economical cycles.
Asset value risk is the risk that is attached to the uncertainty of an aircraft’s future value. PK Air has found a way of predicting the aircraft’s
2
Statistical Aircraft Finance Evaluation
2
future values. This model is also adjusted for the Aircraft Value Cycle and
is in addition affected by a projected inflation path.
The deal structure is the way the deal is set up. This describes the outstanding loan amount or book value over the tenor, fees, rent and interest
rate margins, priority ranking over the collateral (aircraft, deposits, guarantees etc.), rights to upside in the aircraft, and swap breakage exposure.
There are numerous variations of the these structures, and when the counter
party defaults the pay-off function of the lease or the loan will depend on
how the deal was set up. This pay-off function is a function of the asset’s
resale price.
Different deal structure examples are now presented, and the pay-offs are
illustrated under the assumption that the borrower/lessee immediately defaults and that the loans are non-recourse3 .
• Lease of aircraft bought for USD 20mm
• Senior loan USD 20mm
• Shared Pari-Passu loan USD 10mm each
• Senior Shared loan USD 15mm and a Junior Shared loan USD 5mm
First we have the lease contract. If the lessor buys an aircraft for USD 20mm
and leases it to an airline the pay-off function takes the shape of Lease USD
20mm in Figure 1.1. Upon default the lessor repossess the aircraft and may
sell it on the market since they own it. Therefore the lessor can at most
experience a loss of USD 20mm but the upside is unlimited in theory.
A Senior loan is conducted by lending USD 20mm to the airline to finance
their aircraft with the aircraft as collateral. If the airline defaults, they have
to sell the aircraft in the market to repay their debt. The best scenario is
that the lender gets his USD 20mm back, and the worst is loosing all USD
20mm. The lender is not entitled to any upside since they do not own any
aircraft in this deal.
If two lenders enter a shared Pari-Passu4 loan, lending out USD 10mm each
to the airline. This results in the payoff function of the Senior loan being
”halfed”, e.g. if the aircraft is sold for USD 10mm both will experience
losses of USD 5mm.
3
non-recourse means that you can only go after the aircraft to recover your money, not
the owner
4
Pari-Passu is latin and may be translated as ”with equal force”, for a loan it means
that you have equal recovery rights
3
In a shared Senior-Junior loan settlement the pay-off shape changes. If
we have the Senior loan of say USD 15mm and the other lessor has USD
5mm as Junior, we will only experience losses after a sale price of less than
USD 15mm, since the Junior tranche experiences the first loss of USD 5mm.
If we instead have the Junior tranche the pay-off will have the shape of Junior USD 5mm.
There is a vast amount of hybrids of these kind of structures but these
lie outside of the scope of this thesis.
Pay−off Deal Structures
10
Pay−off USD mm
5
0
−5
−10
Lease USD 20mm
Senior Loan USD 20mm
Pari−passu USD 10mm
Senior USD 15mm
Junior USD 5mm
−15
−20
0
5
10
15
20
25
30
Sale Price USD mm
Figure 1.1: Examples of Deal Structure Pay-offs upon default
Another part of the deal structure is the Repayment Schedule. It is the
agreement on how the borrower will repay its debt to the lender, i.e. the
points in time and the magnitude of the cash flows from the borrower to
the lender. As an example a fictitious schedule is shown in Figure 1.2 (additional fees and margins has been left out in this example). The total
outstanding debt is 25 million USD that PK Air has lent to a borrower. To
finance this a lender would typically take a loan of 25 million USD in the
floating rate market, and to hedge themselves they will swap a fixed rate for
floating matching the cash flows. The fixed rate is calculated as an effective
rate so that the net present value of the cash flows is zero, see Section 3.6
4
for an example. The repayment schedule is as follows: the borrower repays
0.5 million USD every 6 months and after 15.5 years he pays the remaining
residual of 10 million USD. PK Air has in turn matched this repayment
setting with their lender.
Fictitious Repayment Schedule
25
Debt [Million USD]
20
15
10
5
0
0
2
4
6
8
10
12
14
Time [years]
Figure 1.2: Fictitious Repayment Schedule
5
16
If PK Air’s borrower defaults or there is an early repayment option of
the loan PK Air will not receive the planned cash flows. Instead they have
to finance their upcoming repayments with the present market setting, that
is with the yield curve at that time. Depending on the yield curve, there
might be a gain or loss from this refinancing procedure when the net present
value and the corresponding fixed rate is calculated. Therefore the Exposure at Default is uncertain and dependent on the propagation of the yield
curve. A fictitious illustration of repayment schedule outcome simulations
with percentiles is shown in Figure 1.3. It shows that the uncertainty is
greatest in the middle of the repayment period if we are at time zero today.
In the beginning and in the end of period the uncertainty diminishes to become certain at the endpoints.
Figure 1.3: Fictitious Repayment Schedule Simulation
At today’s date PK Air uses a model that uses a worst case scenario, the
yield curve simulation decays exponentially with time. PK Air is not pleased
with this way of modeling and wants to improve this simulation. They wish
to have a model that is tractable, has few parameters, is fast to simulate
and it should be influenced by the cycle described in the next Section.
6
1.3
1.3.1
The Aircraft Value Cycle
Introduction
The common view is that the Aircraft business is cyclical. For example the
delivery of aircraft, aircraft orders, air traffic and many other factors vary
cyclically with time. The same thing holds for aircraft values. The outcome
of buying, selling, leasing and/or financing aircraft will be very dependent
on the timing of an investment or divestment compared to the aircraft value
cycle.
1.3.2
Historical evidence
PK Air has built up a database with historical resale prices of aircraft,
which at the moment contains around 4000 sales transactions since 1970.
By looking at specific data for each aircraft type PK Air has been able
to analyze depreciation patterns, cyclicality and volatility statistically. In
Figure 1.4 the resale price of the Boeing 737 300 series expressed in constant
1984 USD value is shown. The pattern around the trend line is cyclical.
Boeing 737−300 Series Resale Price in 1984 $
Boeing 737 resale price
Trend
25
Price
20
15
10
5
1990−01−01
2000−01−01
Date
Figure 1.4: Boeing 737-300 Resale Price in constant 1984 USD over time
PK Air has from their broad study of different aircraft type values found
out that the cyclical variation is almost perfectly correlated in time between
different types.
7
1.3.3
Why this behavior?
Classic economic theory is built upon the interaction between supply and
demand. When the demand for an asset is higher than the supply the price
will rise to find an equilibrium and vice versa. The supply and demand
mechanisms for aircraft seems not to be in equilibrium for most of the time.
The explanation to this is the lag when the aircraft manufacturing industry
tries to adapt to the current market situation. Their pace of manufacturing is directly dependent on the predictions of future traffic made by the
airlines. The manufacturing time is normally 12-18 months. This explains
the adjustment lag; by the time of delivery of the aircraft, the market might
have changed and therefore the predictions and so on. This makes the traffic
very hard to predict in the short term.
The demand for aircraft is estimated by air traffic, which can be expressed
as revenue passengers miles per year. The relationship between production
(revenue passenger miles; RP M ) for one aircraft, load factor LF , block
speed V , utilisation hours U and number of seats S is defined as:
RP M = LF · V · U · S
(1.1)
The time series of the RP M growth are shown in Figure 1.5
RPM Growth
0.2
0.15
0.1
0.05
0
−0.05
1980−01−01
2000−01−01
Date
Figure 1.5: RPM Growth 1968 to 2009
This relationship also holds for airlines and the global air traffic market.
Explanation of the parameters:
1. The load factor has been improved due to management system and
new scheduling techniques. On the one hand a high load is good for
8
the airline’s profit, on the other hand opportunity costs may arise
from high load factors. For example the airline might have to reject
passengers in peak times and vice versa if there is a trough.
2. The block speed was improved greatly during the 1960’s when the
aircraft obtained pressurised cabins, and later by first the turbine engines and then the jet engines. Today the block speed is considered as
constant.
3. Utilisation has improved over the years due to longer range aircraft and
optimized scheduling. There are limiting factors though, in general
people do not fancy flights departing or arriving in the middle of the
night or early morning. On average the utilisation is rising since the
long haul traffic is increasing faster than the short haul traffic. The
reason is that long haul planes spends more hours per day than the
short haul planes.
For the short term airlines are capable to adjust load factor and utilisation
to shifts in demand and supply. There is a however a point in time when
they reach the maximum of these parameters, and they must therefore expand by buying more seats or the other way around if the times are tough.
Fortunately, aircraft are globally traded assets and may therefore easily be
transfered between operators.
1.3.4
Amplitude of Cyclical Swings
In Figure 1.6 the annual growth of physical installed seats (delivered less
scrapped) is shown and the cumulative difference between RP M and Seat
growth adjusted for the trend is shown in Figure 1.7.
We can see the cyclical behavior in the graph which is the driving process for
the aircraft value swings. Amplitudes greater than zero is a measure of the
pent-up relative capacity shortage in the global aircraft fleet, and an amplitude less than zero describes a pent-up surplus. Let us denote this measure
pent-up relative capacity shortage/surplus (PURCS ). In the peaks, airlines
have high load factors and the aircraft is in use many hours per day. There
will be a demand for aircraft so prices will go up. In troughs, the converse
holds.
1.3.5
Prediction of the cycle
To know whether to invest or not we need an prediction about the future.
The (PURCS ) cycle is affected by; traffic growth and seat growth. If we
9
Seat Growth
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1980−01−01
2000−01−01
Date
Figure 1.6: Seat Growth 1968 to 2009
PURCS Cycle
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
1980−01−01
2000−01−01
Date
Figure 1.7: PURCS Cycle 1968 to 2009
10
could predict those for the time horizon of a cyclical swing (approximately
5 years) we would know when the peaks or troughs appear.
Looking at the RPM growth in Figure 1.5 above we could see that the
volatility is high. The notoriously hard predicted GDP Growth drives the
short term variations. For prediction PK Air has two in-house teams, which
consists of members from both the parent company GECAS and PK Air.
The first team is responsible for predicting the RPM Growth by predicting
the parameters in equation (1.1). The second team predicts Seat Growth
that is defined as
Seat
Growth = F + P − C − R
(1.2)
where F is current fleet seats, P is produced seats, C is converted seats
(origins from aircraft being converted from passenger to cargo aircraft (the
converse does not happen)) and R is retired seats.
In the simulations in the model, RPM Growth and Seat Growth are not
simulated independently, instead PK Air starts out in the latest known peak
or trough and simulates the coming peaks and troughs. In the simulation
there are stochastic variables representing the uncertainty in the occurrence
(time) and the level of the next peak or trough. The predictions are revised
and the PURCS Cycle is updated and improved on a quarterly basis.
11
12
Chapter 2
Literature Study
There is an enormous amount of models and research on interest rate modeling. Most of the papers focus on short rate models, e.g. Hull-White, Ho-Lee,
Vasicěk and Heath Jarrow Morton (HJM) and Cox Ingersoll Ross (CIR). All
these models is priced under the risk neutral measure. The parameters in
the models (eg. reversion speed, jump frequencies) implicitly embed, after
being calibrated to market prices, a component related to risk aversion and
therefore contain drift terms different from the objective measure. The drift
terms creates a gap between the risk-neutral and the real world evolution of
the yield curve [17]. Using these short rate models in long horizons results in
evolutions that becomes totally dominated by the no-arbitrage drift terms
and do hardly possess any virtual resemblance to the real-world evolution.
For example Vasicěk and CIR cannot even recover an arbitrary exogenous
yield curve, and the market price of risk is specified (proportional or proportional to the square root of the short rate) have been chosen to allow
analytical tractability.
Another problem with these models is that they are often supposed to be
used with short time horizons such as days or weeks, e.g. for banks that
trade with interest rates. PK Air’s typical deals have an average life of
around 9 years, i.e. the time horizon is much longer than for banks. In
addition their simulations are done in the real world measure, not in a riskneutral measure, and they price the risk at the end of their simulations.
To limit the scope of this study and to adapt to PK Air’s time horizon, I
have chosen in agreement with my academic supervisor and PK Air, not to
look at the short rate models. Instead the focus is on Principal Component
Analysis (PCA) and curve fitting techniques to understand and model the
movements of the yield curve.
Litterman and Scheinkman [11] investigated the common factors that affect the returns on U.S. government bonds and similar securities. They
compared duration analysis with PCA for hedging purposes. The duration
13
concept is used to describe parallel shifts of the yield curve, but in real life,
the yields do not always move in parallel shifts. With PCA they found three
factors affecting the returns of the bonds and that they explained 96 % of
the variation. The factors are referred to as the level, the steepness and the
curvature. The first factor explained 89.5% of the variation, the steepness
8.5% and the curvature 2.0%. They concluded that this PCA representation
is better than traditional duration hedging.
Gloria Soto [21] evaluated the performance and stability of PCA to explain term structure movements. Comparisons were done with typical oneand two-factor interest rate models. She focused on the Spanish government
debt market from January 1992 to December 1999. The result was that the
PCA with with three factors (level, slope, curvature) outperformed the oneand two-factor models. There were however some concerns with the stability
of the PCA over time. The model’s performance deteriorated significantly
when the model is estimated from the most recent data in contrast to longsample data. This points towards the use of factor models with dynamic
volatility structures.
Scherer and Avellaneda [20] used PCA to study the Brady Bond Debt of
Argentina, Brazil, Mexico and Venezuela from 1994 to 1999. They found
that the two first principal components explains up to 90% of the variation.
The analysis was split up into windows to show how the components vary
with time and tried to link those to different economic events. Because of
the short series of data, they looked at daily yield changes, since their results
with weekly or monthly observations were dominated by noise and lacked
structure. For the analysis they looked at absolute yield changes. They
also verified that the results were invariant to valuation basis, that is if they
e.g. used relative changes. Their opinion was that PCA on long observation
windows was best for mature markets with stable economic cycles, which is
compatible with the mathematical definition of stationarity. This was not
the case for this market and they therefore used 120-day windows. They
conclude that there was a high risk for under-estimating the behavior of the
market if considering only a static PCA over the whole period.
Wesley Phoa [16] also compared traditional interest rate management such
as duration management with PCA on the US Treasury market. The PCA
was performed with daily changes in yield on Constant Maturity Treasuries.
He pointed out that the result will only be meaningful if a consistent set of
yields is used and therefore he used Constant Maturity Treasury yields. An
alternative approach was to use historical swap rates, since these are par
yields by definition. The results said that the most important shifts of the
yield curve were the level and the slope. These two shifts can be estimated
quite precisely and robust over time. The curvature shifts tended to be
14
more varying and were highly dependent on the dataset used. When using
shorter term to maturities the variance explained by the level diminished
a little and the slope and curvature increased slightly. The importance of
the three components for bonds was also shown to be similar for all large
countries e.g. the U.S., Germany, France, UK, Australia and Japan. He
also pointed out that some risk factors are ignored with PCA, for example
if there would be a shift in the curve for maturities from 30 to 100 years
relative to shorter yields. This would not show up in the PCA since the
range only goes to 30 years. Another yield curve risk that does not show up
using Constant Maturity Treasuries is the the yield spread between liquid
and illiquid assets. This was a large factor in the US Treasury market in
1998 that was greater than the curvature factor.
Rebonato, Mahal, Joshi, Buchholz and Nyholm [17] evolved a method to
construct a future yield curve over a period of years using a simple semiparametric approach. They used PCA to analyze percentage changes of USD
LIBOR and Swap yields from 29th of September 1993 to 4th of December
2001. They found that the first component explains more than 90% of the
variation. To model future yields they tried different models. The first one,
referred to as the naive simply used historical simulation to evolve future
yields. The drawing of historical yields, was done by randomly choosing a
starting date and then randomly choosing a window of consecutive yields
that should be included. In this manner a yield curve was evolved. This
procedure preserved the co-dependence structure of the original data across
maturities. But they found that the assumption that the increments were
IID did not hold. The method fails to capture the serial autocorrelation
structure and the cross-sectional co-dependence. The difference from IID
behavior was so strong that the procedure was unsatisfactory. To cope with
this they introduced mean-reversion for the shortest and longest maturities and also introduced ”springs” for maturities in between. The springs
were introduced to match the curvatures of the empirical data. The last
refinement consisted of a jump frequency in the sampling windows, that is
a small probability of jumping out of the sampling window and start a new
sampling. By these enhancements they managed to replicate the statistics
of the empirical data very well. They claimed that one possible application
of the model is off-balance sheet transaction, e.g. swaps, which exposes the
counter-party to potential future credit exposure, which is dependent of the
deal will be in the money at the time of a possible default.
Rebonato and Nyholm [18] extends their paper from 2005 by comparing their
semi-parametric approach with a parametric approach for long-horizon yield
curve forecasts. The parametric approach is a model by Bernadell, Coche
and Nyholm [3] that is a three-factor representation of observed yields built
upon the Nelson and Siegel methodology [14]. They found that their semi15
parametric model discovers the same features as the parametric model without the use of Markov chains. The data focused on was US Treasury yields
with maturities; 3 months, 6 months, 1 years, 2 years, 5 years, 10 years, 20
years and 30 years from 1st of January 1986 to 7th of May 2004 at a daily
frequency.
Fiori and Iannotti [7] used Principal Component Analysis and Monte Carlo
Simulation to assess Italian banks interest rate exposure. They looked at the
Euro Area par yield curve with maturities from 1 month to 30 years from
4th of January 1999 to 30th of September 2003. The first three components
explained 95% of the total variation. Instead of simulating from the normal
distribution, which have been proved to be empirical incorrect, they focused
on heavy-tailed distributions. In particular they used an Gaussian kernel
estimator with optimal bandwidth. The distribution functions of these were
obtained by integrating the kernel densities. From these Value-a-Risk calculations were made. They compared the parametric simulation (Normal)
with the non-parametric (Gaussian kernel). The results showed that the
parametric approach has limitations when the interest rates are increasing,
especially when interest rates were low as in December 2001.
Duffee [6] investigated the idiosyncratic variation of Treasury Bill Yields
and found that Treasury bills of three month or less term to maturity exhibits price movements that are idiosyncratic, i.e. they are not related to
changes in other interest rates. He believed that this is caused by an increased market segmentation.
Nelson and Siegel [14] introduced a simple and perspicuous model for the
shapes of the yield curve. They applied it to US Treasury bill and Treasury
bonds yields. The parametric model was successful in representing typical
yield curve shapes: monotonic, humped and S-shaped. They found that the
model explained 96 % of the variation in bill yields across maturities from
1981 to 1983. The model is explained more thoroughly in Section 3.17.
Bank of Canada investigated the Nelson Siegel and Svensson parametric
yield curve models [5]. They investigated the optimization problem, the
robustness and data filtering applied to Canadian Government Securities.
They found that the Svensson model was the best model.
16
Chapter 3
Theoretical Background
3.1
Inflation
Inflation is the rate of change in general prices over time [12]. It can be
expressed in terms of an inflation rate i. The prices 1 year in the future
can on average be expressed as today’s prices multiplied by a factor (1 + i).
The inflation compounds in a similar way to interest rates, that is after k
years at the same inflation rate i, the future prices will be the original prices
times (1 + i)k . In reality the inflation rate changes over time.
Inflation can also be seen as it reduces the purchasing power of money.
A classic example is that a dollar today does not yield as much bread and
milk as a dollar did 10 years ago. Therefore we can think of the prices as
increasing or the value of money as decreasing. If we assume that the inflation rate today is i, then the value of a dollar next year expressed in today’s
purchasing power will be 1/ (1 + i).
When conducting a study over a certain time period, it can be very convenient to express prices in the same kind of dollars. Therefore we consider
constant (real) dollars, which are defined relative to a reference year (typically the starting year). This yields that the dollars will maintain the purchasing power of the dollars for the reference year over time. These dollars
should be distinguished from the actual (nominal) dollars that are used in
daily transactions.
From this we are able to define a new interest rate, namely the real interest
rate, defined as the rate that real dollars increase if they were put into a
bank account that pays nominal interest rate. To illustrate this, think of
putting money into a bank account at time zero and then withdrawing them
17
after a year. The purchasing power of the bank balance can be expressed as
1 + r0 =
1+r
,
1+i
(3.1)
where r is the nominal interest rate, i is the inflation rate and r0 is the real
interest rate. This equation tells us that the money in the bank grows with
1
(1 + r) nominally but it is at the same time deflated by 1+i
. Rearranging
the expression we obtain
r−i
r0 =
.
(3.2)
1+i
For small inflation levels the real rate is approximately equal to the nominal
rate minus the inflation rate.
3.2
Present Value
The idea behind the concept of Present Value (PV) [12] is to describe the
time value of money. If you invest 1 USD at the bank today at a rate rf
(consider as the risk-free rate) you money will be worth 1 · erf ·1 after one
year using continuous compounding.
On the other hand 1 USD recieved one
r
·1
f
year from now is worth 1/ e
USD today which is also referred to as the
present value. The procedure of transforming future cash-flows into today’s
value is called discounting.
3.2.1
Compounding of rates
There are two ways of compounding rates; frequent and continous compounding. If we assume that the risk-free is rf and the interest is compounded at m equally spaced times per year. We do also assume that
the cash flows start out at time zero and at the end of each period, and
that the total number of periods is n. Then we obtain cashflows as follows
(x0 , x1 , ..., xn ) and this yields the following formula for the present value
PV =
n
X
xk
[1 + r/m]k
k=0
(3.3)
If we instead assume that the interest is compounded continuously and that
the cash flows occur at times t0 , t1 , ..., tn . Then the formula boils down to
PV =
n
X
x (tk ) e−rtk
k=0
18
(3.4)
3.3
Forward Rates
Forward rates [12] are interest rates for money to be borrowed between two
future time points, but the terms are agreed upon today. If we for example
consider a 2-year horizon. Assume that the spot rates s1 and s2 are known
and given on a yearly basis. After two years 1 USD would have grown to
(1 + s2 )2 USD in a 2-year account. One other option is to first put the
money in a 1-year account for one year, which will yield (1 + s1 ) after one
year. If we then already have agreed to borrow the money for 1 more year
at a rate f , then we will recieve (1 + s1 ) (1 + f ) at the end of year 2. f
is called the forward rate. These two ways of borrowing money should be
equal if there is no arbitrage and therefore we get the following equation
(1 + s2 )2 = (1 + s1 ) (1 + f )
(3.5)
which can be solved for f
f=
(1 + s2 )2
−1
(1 + s1 )
(3.6)
as can be seen the forward rate is determined by the two spot rates s1 and
s2 .
3.4
Yield
The yield [12] of a bond is the rate of interest that is implied by the payment
structure. More precise it is the interest rate for which the present value of
the bond’s cash flow streams (coupons and face value) is exactly equal to its
current price. The more formal term of yield is yield to maturity (YTM)
to separate it from other yield measures. Yields are always quoted on annual
basis. The yield is exactly the internal rate of return of a bond at its current
price, but yield is the term used in the market. The calculation for the yield,
y, of a bond maturing at time T using continuous compounding is
−y(T −t)
PV = F · e
+
N
X
Ci · e−y(T −ti ) ,
(3.7)
i=1
where F is the face value of the bond, N is the number of coupons and Ci is
the coupon paid at time ti . Equation (3.7) must be solved for y to determine
its yield. It can only be solved by hand for very simple cases, otherwise it
has to be done by an iterative process, which can easily be implemented on
a computer.
19
3.4.1
Yield Curve
The general conditions in the fixed-income securities is closely related to the
yield of a bond [12]. In general all yields move together in this market, but
all yields are not exactly the same. The reason for the variation in yield
over bonds is that they have different credit quality ratings. An AAA-bond
(highest rating) has in general a higher price and thus has a lower yield than
a B-bond with the same maturity. There are however other explanations to
the differences in bond yields. Time to maturity is another factor that affects
the yield of a bond. Intuitively bonds with long time to maturity tend to
offer higher yields than bonds with shorter time to maturity with the same
rating. Therefore a ”normally” shaped yield curve has higher yield with
longer time to maturity. It does in fact happen that bonds with longer time
to maturity have lower yields than bonds with shorter time to maturity,
this is termed inverted yield curve. This shape appears when the shortterm rates increases rapidly, but the investors believe that the increase is
only temporary, and therefore the long-term rates (which is an expectation
about the future) remain close to their former level.
3.5
Interest Rate Swap
A swap [10] is a an agreement between two counter-parties to exchange cash
flows according to a prearranged formula in the future. The first swaps were
settled in 1981 and since then the market has grown fast. Many billions of
dollars of swap contracts are negotiated each year.
The ”plain vanilla” interest rate swap is the most common one. The setup
is that one party, A, agrees to pay another party, B, cash flows of a predetermined fixed rate on a given notional principal at specific points in time
during an interval. A the same time party B pays cash flows to party A,
based on a floating rate on the given notional amount over the same time
period. The currency is the same for the two interest rate cash flows. The
swaps have duration for 2 to 15 years.
Swaps appear because companies have comparative advantages. One company may have an advantage in the fixed rate market and the other in the
floating market. The companies tend to borrow money in the market they
have an comparative advantage. However they might want to match a cash
flow in the other market, this is where the swap is used to transform the
rate. The floating rate used in these agreements is typically the LIBOR
rate, see Section 3.7 for further details.
Financial institutions have a natural advantage funding themselves in the
floating rate market. This implies that a interest rate swap occurs when
20
they want to match for example a deal with an airline who wants to borrow
money with a fixed rate. Then PK Air turns to their lender and matches a
floating rate with a fixed rate that is offered to the airline. All of PK Air’s
interest rate swaps agreements so far has been in US Dollars.
Figure 3.1: Interest Rate Swap
3.6
Swap Breakage
Swap breakage occurs when a swap agreement is terminated early, for PK
Air there could be two causes of this; one is an early repayment and the
other is a default of the counter party. The fixed rates are of course at
the same level throughout the agreement. The floating rate will naturally
change over time. The fixed rate was calculated from the yield curve at time
t0 = 0. On the time of default tdef ault , the yield curve will look different and
the floating rates would imply another fixed rate for the remaining term. In
the present value calculation there is a high probability that the risk-free
rate that is used for discounting has also changed. The present value can be
both positive and negative and of course also zero, resulting in an income,
cost or zero result breakage.
3.6.1
Example of Swap Breakage
Say that we want to swap a floating rate for a fixed rate for a time interval
of 5 years and that the swaps are made on a yearly basis. The swap is on
21
a notional amount, N , of 50 000 USD. Assume that today’s yield curve has
the following setting:
Table 3.1: Fictional Yield Curve
Term to maturity
1 yr
2 yr
3 yr
4 yr
rate
0.0058 0.0124 0.0186 0.0233
5yr
0.0270
where the rates are compounded on a yearly basis. From this we can calculate the implied forward rate fij between year i and j. For the second year
this is
(1 + 0.0124)2 = (1 + 0.0058) (1 + f12 )
if solved for f12
f12 =
(1 + 0.0124)2
− 1 = 0.0190
(1 + 0.0058)
in the same manner we can calculate all forward rates.
Table 3.2: Implied Forward Rates
interval
0-1 yr 1-2 yr 2-3 yr 3-4 yr
forward rate 0.0058 0.0190 0.0312 0.0375
4-5 yr
0.0419
To calculate the fixed rate, we calculate the Present Value (PV) in thousands USD of the cashflow streams and we set the PV to zero and solve for
rf ixed
50 (0.0058 − rf ixed ) e−0.0058·1 + ... + (0.0419 − rf ixed ) e−0.0270·5 = 0
0.0058 · e−0.0058·1 + ... + 0.0419 · e−0.0270·5
= 0.0265
e−0.0058·1 + ... + e−0.0270·5
If we now assume that the agreement is terminated after 1 year and we have
the following yield curve at that time
⇒ rf ixed =
Table 3.3: New Fictional Yield Curve
Term to maturity
1 yr
2 yr
3 yr
4 yr
rate
0.0122 0.0152 0.0223 0.035
22
As before we back out the implied forward rates
Table 3.4: New implied Forward Rates
interval
0-1 yr 1-2 yr 2-3 yr 3-4 yr
forward rate 0.0122 0.0182 0.0367 0.0740
From this we obtain the following Present Value
P V = (0.0122 − 0.0265) · e−0.0122·1 + ... + (0.0740 − 0.0265) · e−0.035·4 · 50
= +1.7058
If the counter-party defaults we will have a Swap Breakage Gain of 1705.8
USD.
3.7
LIBOR rate
The information in this section was found at one of the British Bankers
Association (BBA) webpages [2].
3.7.1
Historical Background
In the early 1980’s an increasing number of banks in the London market
were actively trading new instruments as currency options and interest rate
swaps. Most of the banks found these new instruments attractive but they
were at the same time bothered about that the underlying rates that had to
be agreed on before entering the contract. The BBA were therefore asked
by the banks they represented to create a uniform measure for the market
and produce a benchmark index. The concept was that banks could now
reference their contracts against a standard rate which removed the negotiation of the underlying rate. This improved the generation of instruments.
In 1984 BBA cooperated with for example the Bank of England and others
which resulted in the BBAIRS, that is the BBA standard for Interest Swap
rates.
In this standard the fixing of BBA Interest Settlement rates, the predecessor
of BBA-LIBOR, was included. BBAIRS became standard market practice
23
the 2nd September 1985. The first BBA-LIBOR rates were given in 1986
and in three currencies; US Dollars, Japanese Yen and Sterling. Today it is
calculated in 10 currencies for 15 maturities.
Since the BBA-LIBOR was introduced there has been two significant changes.
Firstly in 1998 the question was changed from ”At what rate do you think
interbank term deposits will be offered by one prime bank to another prime
bank for a reasonable market size today at 11 am?”. The new question,
which is still used today is ”At what rate could you borrow funds, were you
to do so by asking for and then accepting inter-bank offers in a reasonable
market size just prior to 11 am?”.
Secondly in 1999 with introduction of the Euro on the 1st of January 1999
the number of currencies BBA-LIBOR was calculated for obviously diminished.
3.7.2
BBA-LIBOR
BBA-LIBOR stands for London InterBank Offered Rate. The rate is derived
for 10 currencies with 15 maturities for each, the shortest being overnight
and the longest being 12 months. The BBA-LIBOR should be interpreted
as a benchmark, it provides you with an idea of what the average rate of
a prominent bank, for a specific currency, can get unsecured funding for a
certain term in the specific currency. In other words, it yields the lowest
real-world cost of unsecured funding in the London market.
The rates are derived by a calculation based on submissions from a panel,
consisting of the most active banks in the specific currency.
3.7.3
Definition
It is important to know that BBA-LIBOR is based upon the offered rate,
and not the bid rate. Each contributing bank is ask to base their BBALIBOR submission on the the question mentioned above: ”At what rate
could you borrow funds, were you to do so by asking for and then accepting
inter-bank offers in a reasonable market size just prior to 11 am?”. In this
way the submissions are based upon the lowest perceived rate that a bank
that participates in a certain currency panel could go into the inter-bank
money market and get sizable funding for a certain maturity.
The BBA-LIBOR is not derived from actual transactions, it would be hard
to have this as an requirement since not all banks require funds in marketable size each day for the currencies and maturities they quote. This
should not be interpreted as that the rates does not reflect the true cost of
24
interbank funding. The banks are aware of their credit and liquidity risk
profiles for the rates it has dealt with, and are therefore able to build a curve
that accurately predicts the correct rates for currencies or maturities which
it has not quoted at the moment.
All BBA-LIBOR rates are quoted as annualized interest rates, which is a
market convention. For an example say that the overnight rate is given as
2 %, this does not imply that that the overnight loan interest rate is 2 %.
Instead it means that the annual loan interest rate is 2 %.
3.7.4
Applications
BBA-LIBOR is the primary benchmark for short term interest rates in the
world. The rate is also used as an indicator of the strain in money markets
and it is also often used as a measure of the market’s expectations of future
central banks interest rates. Different studies show that approximately 350
trillion USD of swaps and 10 trillion USD of loans are indexed to BBALIBOR. It is the foundation for settling interest rate contracts on most of
the world’s major futures and options exchanges.
3.7.5
Selection of panels
The contributing banks are chosen for currency panels with the goal of
reflecting the market balance for a specific currency based upon these three
guidelines
1. Scale of market activity
2. Credit rating
3. Perceived expertise in the currency concerned
Each of the 10 panels, with 8 to 16 contributors, is chosen by the independent
Foreign Exchange and Money Markets Committee (FX & MM Committee)
to obtain the best way of representing the activity in the London money
market for a certain currency. Therefore BBA-LIBOR submissions from
panel members will on average be the lowest interbank unsecured loan offers from the ones available on the money market.
The FX & MM Committee evaluates each panel every year, based upon
a review from BBA of the contributors. The review evaluates each bank
according to the criteria listed above. The review is not limited to only
present contributors as any Bank can submit itself to the evaluation process
for any currency.
25
3.7.6
Calculation
The calculations are made by Thomson Reuters. They analyze the data
from the banks and calculate the rates according to the definitions given
by FX & MM Committee. The whole procedure is supervised by BBA. In
the contributing banks every cash desk has an application from Thomson
Reuters installed. Each morning the currency dealer takes their own rates
of the day and puts them in to the application between 11.00 and 11.20.
The banks are not able to see the other banks rates until they are submitted
by Thomson Reuters. The data is also submitted by nine other data vendors.
All the BBA-LIBOR rates from Thomson Reuters are calculated in the
same way using an trimmed arithmetic mean. The contributors are ranked
by Thomson Reuters and sorted in descending order. The trimming is done
by dropping the top and bottom quartiles. The remaining 50 % of the data
are then averaged to get the BBA-LIBOR quote. The reason to drop the top
and bottom quartiles is to improve the accuracy of the quotes by removing
outliers. By dropping these the possibility for any individual contributor to
influence the quote is removed.
3.8
U.S. Interest Rate Swaps
These rates can be used as a proxy for the continuation of the LIBOR
rates. These are quoted as the fixed rates swapped for 3 month LIBOR
semiannually.
3.9
3.9.1
US Government Debt
Treasury Bills
The U.S. government offers debt obligations called Treasury bills [19] , also
known as T-bills. Of the total U.S. government marketable debt the T-bills
occupy approximately one-fourth. In the middle of 1995, there was around
three-quarters of a trillion dollars worth of bills outstanding. Since the bills
are liabilities of the government, these obligations are considered as default
free. The secondary market for these are one of the most active, characterized by its low bid-ask spread and extremely high liquidity.
The T-bills are issued at discount from the face value and have no coupons
or stated interest. The earnings for the holder is simply the difference between the discount issue price and the face value, which is paid out by the
Treasury at time of maturity. The discount is determined in an auction,
where new bills are offered to dealers and other investors. The auction is
conducted by the Federal Reserve Bank of New York acting on behalf of
26
the U.S. Treasury. The bills are sold to those who offer the highest price,
and this auction ensures that the resulting interest costs of the Treasury
is minimized. The discount is related to the term of the bond and what
investors believe about the market future.
T-bills are offered in the following maturities: 91-day (3-month), 182-day(6month) and 364-day(1-year). The three- and six-month bills are auctioned
every week, and the one year every fourth week to meet the huge demand for
funds from the U.S. government of refinancing an outstanding debt. Since
1993, the bills that have been issued in multiples of 1000 USD, with 10000
USD as the smallest amount. The amount which has the lowest commission rates is the one known as a round lot and is 5 million USD. These
are normally traded by large market participants. The commission rates is
generally in between 12.50 to 25 USD per 1 million USD and is affected by
the maturity of the bill, three-months have the lowest commission.
No certificates or papers of the debt is issued when the T-bills are issued.
The claim is only registered in the computer system of the Federal Reserve.
Due to their low risk and short maturity, T-bills are very popular instruments for market participants. The range of holders goes from individuals
to governments and everything in between. For individuals and commercial
investors the T-bills is interesting since they are relieved from state and local
taxes. Foreign banks are large holder of bills, their interest is mostly based
on the safe return of the bills. For the U.S. government the T-bills are an
important way to raise funds to finance the U.S. debt outstanding. The
Federal Reserve does also hold a lot of bills to be able to use its monetary
instruments to influence the economy.
3.9.2
Treasury Bonds and Treasury Notes
Contrary to the T-bills, the Treasury bonds (T-bonds) and Treasury notes
(T-notes) [19] are interest-bearing securities with maturities greater than
one year. The difference between notes and bonds is that they are issued
in different maturities. Notes are issued with maturities of 2, 3, 4, 5, 7 and
10 years and bonds with maturities greater then 10 years, e.g. 30 years.
The primary market for these securities is a bit complicated. Almost all
Treasury debt that is negotiable is offered at auctions on a yield basis. The
speculators submit sealed bids with the lowest yield to maturity that they
would accept, instead of regular bidding auction. The bids are conducted
at local Federal Reserve Banks and they are opened at a prespecified time
in the future. The bids specify the volume and the yield (in percent in two
decimal places). There is also a possibility for small investors to place a
noncompetitive bid, which means that they will pay a price equal to the
mean of the accepted bids. The Treasury itself does intervene the process
27
by setting ”stop-out” levels, which is the highest level that they are willing
to issue debt. At the prespecified times the average level of the accepted
bids are calculated. Then the Treasury sets a coupon on the debt closest to
1/8 of 1 percent, where the level will yield an average price of the successful
bids equal to 100 USD (per 100 USD of face value) or lower.
3.10
Smoothing Spline
If the available data set is very noisy, e.g. daily yields of a certain maturity,
it might be a good idea to fit your data with a smoothing spline [13]. A
smoothing spline, s, is constructed for a given smoothing level p and weights
wi . The objective of the spline is to minimize
p
X
Z 2
wi (yi − s (xi )) + (1 − p)
i
d2 s
dx2
2
dx
if the weights, wi , are unspecified all points are assigned a weight of 1. The
smoothing level, p, is defined between 0 and 1. The lower limit produces a
least-square straight line fit to the data. If you set it to the upper boundary
it produces a cubic-spline interpolant. An application of this procedure is
given in Section 5.1.3 later on.
3.11
Correlation
To get an idea of what correlation is, consider the joint density function of
two random variables [1]. This density could be said to describe a mountainlike shape. If the mountain is symmetric around the two axes of the random
variables, little information about one variable is obtained by knowing the
value of the other variable, i.e. the correlation is low between the variables.
On the other hand, if the shape of the joint density function is ridge-like
between the axes, the correlation will be high.
To get a feeling for the correlation, scatter plots is useful. A scatter plot
is a plot of two synchronous returns of financial time series against each
other, e.g. gold returns against crude oil. A low correlation corresponds to
symmetrical dispersed scatter plot, i.e. a high value of one variable does
not imply a high/low value of the other. A high correlation is described
by a ridge between the axes, if it has negative the slope the correlation is
negative, and vice versa.
Correlation measures the co-movements between two return series. Strong
positive correlation says that an upward movement in one return series is accompanied by an upward movement in the other series. For strong negative
28
correlation an upward movement in one series corresponds to a downward
movement in the other series.
The covariance measure can be used to measure the co-movements of two
random variables X and Y , it is defined as
Cov (X, Y ) = E [(X − µX ) (Y − µY )]
(3.8)
where µX and µY are the means of the random variables. The drawback
with this measure is that it is not only determined by the co-movements
of the returns but also by the size of them, that is in general monthly
returns will have higher covariance than daily returns. Since covariance is
not independent of the units of measurement, it is not suitable to make
comparisons. To cope with this the correlation measure is introduced. It is
a standardized form of the covariance that it is independent of the units of
measurement. For two random variables it is standardized by dividing the
covariance by the product of their standard deviations, that is
Cov (X, Y )
Corr (X, Y ) = p
V (X) · V (Y )
(3.9)
where V (X) and V (Y ) are the variances. This standardization procedure
will yields a correlation value of -1 to +1, where the first refers to perfect
negative correlation and the latter to perfect positive correlation. If the two
random variables are statistically independent their correlation coefficient
should be insignificantly different from zero, and the variables are referred
to as orthogonal. Notice that the converse is not necessarily true, if two
random variables are orthogonal does not imply independence (they could
have zero covariance but still be related by the higher moments of their joint
density function).
3.11.1
Cross-Correlation
The autocorrelation function could be used to determine a sequence’s structure in the time domain. Cross-correlation uses the same concept but instead
of comparing a time shifted version of the signal with itself, it compares to
different sequences with each other [9]. The cross-correlation function of two
sequences {Xt } and {Yt } is defined as
T
X
1
Xt Yt+h
T →∞ 2T + 1
φX,Y (h) = E (Xt Yt+h ) = lim
(3.10)
t=−T
The function is a statistical comparison of two sequences as a function of the
time shift between them. It is very useful as a practical tool for determining
timing differences between the sequences.
29
3.11.2
Cross-Correlation Coefficient
The Cross-Correlation Coefficient is closely linked to the usual correlation
coefficient and is defined as
φX,Y (h)
`X,Y (h) = p
φX,X (h = 0) φY,Y (h = 0)
(3.11)
as with the traditional correlation coefficient, the value of the cross-correlation
coefficient lies between -1 and 1. It can be used to determine for which lag
h to sequences obtain the highest correlation.
3.12
Least Squares Method
Let x1 ,...,xn be outcomes of independent stochastic variables X1 ,...Xn . The
means of the stochastic variables are known except for an unknown parameter. Therefore it is assumed that E (Xi ) = µi (θ) for i = 1, 2, ..., n where
µ1 , µ2 , ..., µn are known and θ is an unknown parameter with parameter
space ΩΘ . This implies that Xi = µi (θ) + i , i.e. Xi = ”known function
of θ” + ”error”. These errors are assumed to have zero means. In general
their variances are assumed to be identical. Let
Q (θ) =
n
X
[xi − µi (θ)]2
(3.12)
i=1
be the sum of the observations deviations from the µi (θ)’s. The expression
between the hard brackets are recognized as the error of observation i if θ
is the correct parameter value. The method is to use the θ-value that minimizes the sum of squares as θ-estimate. To find this Least Squares Estimate
of θ [4] the derivative dQ (θ) /dQ is calculated and set equal to zero to find
the minimum.
A more general method can be applied if the distribution contains k unknown parameters θ1 , θ2 , ..., θk . The expression for Q is now a function of
these parameters
Q (θ1 , θ2 , ..., θk ) =
n
X
wi [xi − µi (θ1 , θ2 , ..., θk )]2
(3.13)
i=1
where w1 , ..., wn are weights that could be assigned to the observations. In
general these are set to 1, which corresponds to equal uncertainty for all
data. The Least Square Estimate is as before determined by minimization,
e.g. by setting the partial derivatives of Q to zero and solving for θi .
30
3.13
Maximum Likelihood Method
The idea behind the Maximum Likelihood Method is to as an estimate of
θ use the value that makes our sample data as likely as possible [4]. The
value of θ will therefore naturally depend on our sample data and will be
∗ of θ.
the Maximum Likelihood Estimate, θobs
Let x1 ,...,xn be outcomes of independent stochastic variables X1 ,...Xn that
has a distribution that depends on an unknown parameter θ with parameter
space Ωθ . In most cases X1 ,...Xn are assumed to be independent and that
they have the same distribution, but the method also works without these
assumptions.
If X is a continuous stochastic variable it has density function f (x; θ), and
in the discrete case it has probability mass function p (x; θ). If the sample is from a discrete distribution and all the Xi0 s are independent, will the
probability of obtaining P (X1 = x1 , ..., Xn = xn ) be
P (X1 = x1 , ..., Xn = xn ) = P (X1 = x1 ) · · · P (X1 = x1 )
= pX1 (x1 ; θ) pX2 (x2 ; θ) · · · pXn (xn ; θ)
(3.14)
(3.15)
The Likelihood Function is defined as
P (X1 = x1 , ..., Xn = xn ; θ) (discrete)
L (θ) =
fX1 ,...fXn (x1 , ..., Xn = xn ; θ) continuous
The idea behind the Maximum Likelihood method is to let the argument θ
take all values in Ωθ and find the value of θ that maximizes the function.
∗ and is the estimate. In the discrete case the idea beThis value is called θobs
hind the method is to maximize the probability of obtaining the sample data.
When maximizing L (θ) it is often convenient to maximize lnL (θ). Since
the logarithm is a monotonic increasing function will both L (θ) and lnL (θ)
have maximums at the same point. The Likelihood function is normally a
product and therefore the logarithm yields a sum which facilitates the maximization which is done by taking the derivative and solving for it equal to
zero.
3.14
Linear Regression
Linear regression models are formulated in this way
Y = β1 X1 + β2 X2 + ... + βk Xk
(3.16)
On the left side is the dependent variable, Y , and on the right there are
the independent variables X1 , X2, ..., Xk [1]. The independent variables can
31
also be called explanatory variables. The coefficients β1 , β2 , ..., βk are model
parameters and they measure the influence of its corresponding independent
variable on Y . X1 is in general assumed to be equal to 1 so the model
includes a constant β1 . The goal of the regression is to find estimates of the
true parameter values and predictions of the dependent variable by using
historical data on the dependent and independent variables.
3.14.1
The Simple Linear Model
The simplest case is when k = 2 and X1 = 1, i.e. we have a constant for
all t. For notational purposes the constant is set to α (interpreted as the
intercept with the vertical axis), and β2 is denote by β (the slope of the line)
and X2 is denoted X. This results in the following setting
Yt = α + βXt
(3.17)
To see the relationship between the data all pairs of (Xt , Yt ) is plotted against
each other in a scatter plot. All the points will not lie along a straight line
so an error process is introduced to the equation
Yt = α + βXt + t
(3.18)
An estimated straight line through the scatter plot yield a predicted or fitted
value of Yt for each Xt from
Ŷt = α̂ + β̂Xt
(3.19)
where α̂ and β̂ are the estimates of the intercept and the slope. The difference between the real value of Y and the fitted value Y at time t is denoted
t and is called the residual at time t, i.e. t = Yt − Ŷt . So each real data
point Yt is described by
Ŷt = α̂ + β̂Xt + t
(3.20)
The estimates are obtained by minimizing the sum of the squares of the
residuals, which is known as ordinary least squares (OLS) criterion. OLS is
an unbiased estimation.
The OLS estimates for the Simple Linear Model are given by
X Xt − X̄ Yt − Ȳ
β̂ =
2
P
t
t Xt − X̄
α̂ = Ȳ − β̂ X̄
where Ȳ and X̄ denotes the sample means of X and Y .
32
(3.21)
(3.22)
3.14.2
Multivariate Models
In the multivariate case the model takes the form
Yt = β1 X1t + β2 X2t + ... + βk Xkt + t
(3.23)
and assuming that the model has a constant term yields X1t = 1 for all t =
1, ..., T . There will be T equations with k unknown parameters. To simplify
things the model can be written in matrix notation. Let the dependent
variable column vector be y = (Y1 , Y2 , ..., YT )0 and put the independent
variables into a matrix X where the jth column of X corresponds to the
data on Xj . The first column of X will be a column of 1s if there is a constant
in the model. Denote the vector of true parameters β = (β1 , β2 , ..., βk )0 and
let = (1 , 2 , ..., T )0 be the vector of error terms. The representation of the
model then boils down to
Y = Xβ + (3.24)
The matrix form of the OLS estimators of β is given by
−1 0
β̂ = X0 X
Xy
3.14.3
(3.25)
Goodness of Fit
There are numerous ways of measuring how well the model describes the
data. Some measures are presented below.
3.14.3.1
R2 and adjusted R2
R2 is the coefficient of determination of a model. It is a product of the
concept of analysis of variance (ANOVA). It has three key metrics;the total
sum of squares, T SS = (y − ȳ)T (y − ȳ), the explained sum of squares,
ESS = (ŷ − ȳ)T (ŷ − ȳ) and the residual sum of squares, RSS = T . R2
is the proportion of the total sum of squares that is explained by the model
T SS
RSS
R2 =
=1−
(3.26)
ESS
T SS
From this measure it is possible to tell how much of the variation in y that
is explained by βX.
Adjusted R2 takes into account the number of explanatory variables in the
2 increases only if the new term improves the
model. Different from R2 , Radj
model more than expected by chance. The measure can be negative and it
is less or equal to R2 . The definition is
2
Radj
=1−
RSS/ (T − k − 2)
T SS/ (T − 1)
(3.27)
where T is the sample size and k is the number of explanatory variables.
2 is in general used for determining if an explanatory variable improves
Radj
the model
33
3.15
Principal Component Analysis
This section follows the derivation in the book by Alexander [1]. Before the
analysis can be done, the data must be stationary. This is done by taking
the absolute difference from day to day in the T × K rate data matrix X.
Then you normalize your data matrix X by subtracting each column with
its mean µ and standard deviation σ to obtain a new matrix X∗ with mean
0 and standard deviation 1.
Principal Components Analysis (PCA) starts out from the symmetric correlations matrix of the variables in X∗
V=
X∗T · X∗
T
(3.28)
the components are found by calculating the eigenvalues and eigenvectors of
the correlation matrix V. It will be shown that the principal components can
be described as a linear combination of these columns. The components will
be orthogonal to each other since they are eigenvectors. The first principal
component will describe the most of the variance in X∗ , the second the
second most and so forth. This is achieved by choosing the weights from the
k × k eigenvector matrix of V in the following way
VW = WΛ
(3.29)
where Λ is the k × k diagonal matrix of eigenvalues from V. Next the
columns of W is ordered according to the size of their corresponding eigenvalue. Therefore if wij for i, j = 1, ...k, then the mth column of W,
wm = (w1m , ..., w1k ) is the k × 1 eigenvector that corresponds to eigenvalue λm . The columns of W is ordered so that λ1 > λ2 > ... > λk .
The mth principal component is defined in the following way
Pm = w1m · X1∗ + w12 · X2∗ + ...w1k · Xk∗
(3.30)
where Xi∗ is the ith column of the normalized matrix X∗ . This can be
written as
Pm = X∗ · wm
(3.31)
The complete T × m matrix of principal components can thus be expressed
as
P = X∗ · W
(3.32)
To understand that this leads to orthogonal (uncorrelated) components,
notice that
0
PP = WT X∗ T X∗ W = {X∗ T · X∗ = T V}
= T WT WΛ = {W
is orthogonal
WT = W−1 }
= T Λ ,where Λ is the diagonal matrix of eigenvalues
34
Therefore the vectors of PT P are uncorrelated.
Each principal components variance is determined by its corresponding eigenvalues proportion of the total variation. For example the variation in X∗
explained by Pm is given by
λm
λm
=
Σλii
k
where k is the number of variables in the system.
(3.33)
If the components in the original system is highly correlated the first eigenvalue will be much greater than the others, i.e. the first principal component explains the most of the variation. Equation (3.32) can be expressed
0
0
as X∗ = P · W because W = W−1 , in vector form
Xi = wi1 P1 + wi2 P2 + ... + wik Pk
(3.34)
It is therefore possible to represent each input data vector as a linear combination of the principal components. This is known as the Principal Component Representation of the original variables. It is often enough to only
include the first few principal components, since these explain almost all the
variation.
3.16
Ornstein-Uhlenbeck process
An Ornstein-Uhlenbeck process is a mean-reverting process given by the
stochastic differential equation
dXt = κ (θ − Xt ) dt + σdWt
(3.35)
where κ, θ and σ are parameters and Wt is a standard Brownian Motion. κ
is the speed of mean-reversion and θ is the mean and σ is the volatility.
To solve the equation we introduce a change of variable, letting Yt = Xt − θ.
Then Yt satisfies the following Stochastic Differential Equation
dYt = dXt = −κYt dt + σdWt
(3.36)
In equation 3.36 the process dYt has a drift towards zero at an exponential
rate κ. Therefore a change of variable is motivated to remove the drift. This
is done by
Yt = e−κt Zt ⇔ Zt = eκt Yt ,
(3.37)
Applying Itô’s formula to equation 3.37 yields
dZt = κeκt Yt dt + eκt dYt
κt
(3.38)
κt
(3.39)
= 0dt + σe dWt
(3.40)
= κe Yt dt + e (−κYt dt + σdWt )
κt
35
The solution is obtained by integrating from s to t
Z t
eκu dWu
Zt = Zs + σ
(3.41)
s
Retransforming the change of variables yields
Yt = e
−κt
Zt = e
−κ(t−s)
−κt
t
Z
eκu dWu ,
(3.42)
e−κ(t−u) dWu .
(3.43)
Ys + σe
s
and finally
Xt = Yt + θ = θ + e
−κ(t−s)
Z
(Xs − θ) + σ
t
s
3.17
Nelson-Siegel’s Yield Curve Model
Nelson and Siegel formulated their model to fit the shape of the yield curve
in the following way


m
m
1 − e− τ
1 − e− τ
m
y (m) = β0 + β1
+ β2 
− e− τ 
(3.44)
m/τ
m/τ
where m is term to maturity, β0 is the long term component, β1 is the short
term component, β2 is the medium term component. The long term component is a constant throughout the maturity, the medium component starts
out at zero and decays to zero and is therefore only affecting the medium
term. The short term component has the fastest decay monotonically to
zero. τ determines the if the fit will be best at short or long maturities and
the location of the maximum of the medium component β2 . A graphical
illustration is shown in Figure 3.2.
Some constrains has to be imposed on the model to obtain plausible
results when fitting it to empirical yields. First β0 , the long term component
has to be greater than zero. Secondly, β0 + β1 has to be greater than
zero to guarantee non-negative yields at the short end. Thirdly, the decay
parameter, τ has to lie in the maturity spectrum, i.e. be greater than zero.
36
Influence of Nelson−Siegel Components with τ = 2
1
0.8
β0
0.6
β
1
β2
0.4
0.2
0
0
5
10
15
20
25
30
Term to Maturity
Figure 3.2: Nelson-Siegel components influence
37
38
Chapter 4
Data Selection
The floating rate that PK Air uses as a reference is the LIBOR rate, this
data was obtained from Bloomberg. Since the LIBOR rates longest maturity is 12 months, US Swap rates have been used as a proxy for longer time
to maturity. The idea was to look at the most homogeneous data sets. The
time series of data were available in the following setting for daily quotes:
Table 4.1: LIBOR and US Swap Time Series 1988-2009
LIBOR
1 m 2 m 3 m 6 m 12 m
USSWAPS 2 yr 3 yr 4 yr 5 yr 7 yr
10 yr
Since the time series only go back to 1988, there was a need for time series
who stretched further back in time. Therefore US Government Securities
yields were also studied, specifically Treasuries with constant time to maturity [15]. The longest homogeneous time series were available from 2nd of
January 1962 and were composed in this way and were available in daily,
weekly and monthly quotes.
Table 4.2: US Treasuries Constant Time to Maturity Series 1962-2009
Treasury 1 yr 3 yr 5 yr 10 yr
Another setting with more maturities were available from 1982
Table 4.3: US Treasuries Constant Time to Maturity Series 1982-2009
Treasury 3 m 6 m 1 yr 3 yr 5 yr 10 yr
39
40
Chapter 5
Results
5.1
5.1.1
Principal Component Analysis
LIBOR and Swaps
Starting out with the LIBOR and Swaps rates some problems appeared with
the raw data. It turned out that the rates had sometimes been quoted on
different days, typically one rate was missing out on a certain day. To deal
with this problem interpolation was applied to obtain homogeneous data
sets. It was now possible to construct a yield surface
Figure 5.1: Nominal Yield Surface LIBOR and Swaps 1988 to 2009
41
Secondly the rates were transformed to real rates using the relation (3.1).
The US inflation data was given on a monthly basis [8] and was applied to
the rates that corresponded to that month. This yielded a real yield surface
as can bee seen the yield surface shape changes.
Figure 5.2: Real Yield Surface LIBOR and Swaps 1988 to 2009
Principal Component Analysis has an application to term structures [1].
Term structures are special because they yield an ordering of the system
that gives an intuitive interpretation of the principal components as shall
be seen. Before performing the PCA it is important that the time series
are stationary, the yields are in general not stationary , therefore absolute
differences of the yield data were considered, X, as shown in Figure 5.3. As
a last step the differences are normalized, that is subtracting the mean and
dividing by the standard deviation for each of the columns to obtain a new
matrix X∗ , a graphical illustration of this for 1 month LIBOR can be seen
in Figure 5.4.
42
1 month nominal LIBOR absolute changes
0.015
0.01
0.005
0
−0.005
−0.01
−0.015
1990−01−01
2000−01−01
date
Figure 5.3: 1 month nominal LIBOR absolute changes
1 month nominal LIBOR normalized absolute changes
20
15
10
5
0
−5
−10
−15
−20
1990−01−01
2000−01−01
date
Figure 5.4: 1 month nominal LIBOR normalized absolute changes
43
From the normalized absolute changes the correlation matrix was obtained
Table 5.1: Correlation Matrix LIBOR and Swaps
1 m
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1 m
1
0.620
0.555
0.463
0.377
0.112
0.108
0.105
0.089
0.086
0.087
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1
0.648
0.556
0.439
0.129
0.105
0.096
0.091
0.089
0.083
1
0.741
0.624
0.196
0.163
0.148
0.143
0.139
0.125
1
0.757
0.229
0.205
0.186
0.179
0.171
0.160
1
0.273
0.249
0.228
0.222
0.212
0.202
1
0.934
0.899
0.887
0.858
0.7924
1
0.935
0.928
0.906
0.842
1
0.936
0.918
0.860
1
0.957
0.916
1
0.936
1
The correlations shows typical yield curve behavior. The correlation diminishes with the spread between the maturities. Another observation is that
the longer term to maturity, the higher the correlation with the neighboring
maturities than for shorter maturities.
Table 5.2: Eigenvalues of Correlation Matrix LIBOR and Swaps
Principal Component Eigenvalue Cumulative R2
P1
5.8070
52.8
P2
3.0414
80.5
P3
0.7745
87.5
P4
0.3842
91.0
The first three components, which are referred to as the level, the slope and
the curvature explains more than 87% of the variation. The table does also
show that the first component corresponds to the largest eigenvalue. Figure 5.5 shows a graphical illustration of the first three principal components
(PCs).
A hump occurs for the shorter maturities, this could be explained as shall
be seen later on by the inclusion of the shorter maturities. The first PC, the
level, is approximately constant over the maturities. The second PC, the
slope, approximately raises the yield for maturities up to 2 years and lowers
it for longer. The third PC, the curvature, approximately raises the yield
for all maturities except the interval 3 months to 3 years.
5.1.1.1
Different Valuation Basis
What happens if the valuation basis is changed from absolute returns to log
returns? Log returns is defined as
Xn = logSn − logSn−1
44
(5.1)
Principal Component Analysis
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.5: Principal Components nominal LIBOR and Swaps
where Si is the yield at day i. The answer is that the results are almost the
same as shown in Figure 5.6 and Table 5.3
45
Principal Component Analysis
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.6: Principal Components nominal log LIBOR and Swaps
Table 5.3: Variance Explained log LIBOR and Swaps 1988-2009
Component Cumulative R2
P1
54.03
P2
84.11
P3
90.79
P4
93.68
The results are therefore indifferent of the valuation basis, the small
differences can be explained by numerical errors in MATLAB.
5.1.1.2
Weekly data
The same procedure was done for weekly data on the yields of LIBOR and
US Swaps. The Correlation Matrix changed as can be seen in Table 5.4.
46
Table 5.4: Correlation Matrix LIBOR and Swaps Weekly
1 m
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
5.1.2
5.1.2.1
1 m
1
0.679
0.734
0.637
0.469
0.310
0.274
0.257
0.229
0.209
0.191
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1
0.839
0.753
0.577
0.346
0.303
0.284
0.271
0.247
0.222
1
0.857
0.726
0.442
0.394
0.365
0.343
0.323
0.311
1
0.824
0.591
0.537
0.502
0.482
0.447
0.423
1
0.641
0.598
0.562
0.547
0.515
0.494
1
0.979
0.941
0.926
0.890
0.873
1
0.968
0.963
0.938
0.914
1
0.964
0.946
0.922
1
0.976
0.951
1
0.962
1
US Treasuries Constant Maturity
Daily Data
The Treasury yield curve go back from 1962, but from then data are only
available for maturities from 1 to 10 years. This longest yield surface is
shown in Figure 5.7.
Figure 5.7: T-notes 62-09 Nominal Yield Surface
47
If the whole period is analyzed at once, the following results are obtained
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.8: Principal Components Nominal T-notes 62-09
48
Table 5.5: Variance Explained T-Notes 1962-2009
Component Cumulative R2
P1
89.05
P2
96.43
P3
98.89
The level explains almost all the variation.
The time interval between 1962 to 2009 was also split into smaller windows
of 5 years. The visual results of how the principal components propagates
over time can be seen in Appendix A. The total variance explained is shown
in Table 5.6.
Table 5.6: Variance Explained T-notes 1962-2009 5 year windows
Cum. R2
P1
P2
P3
62-67
80.66
91.99
96.91
67-72
86.94
94.81
98.42
72-77
81.28
91.95
97.50
77-82
91.26
96.61
98.82
82-87
91.46
96.90
98.93
87-92
91.37
97.76
99.25
92-97
92.66
98.02
99.40
97-02
89.21
97.15
99.20
02-07
89.81
97.52
99.48
07-09
84.48
96.73
99.43
From the table is clear that the level explains more than 80 % of the total
variation and all three components explains more than 96 %. This implies
that when leaving out the shorter term to maturities the explanatory power
increases. From the graphical illustrations of the components the ”hump”
that occurred with the LIBOR and US Swaps has now disappeared. The
explanation is that the shorter term to maturities are not included in the
T-notes and that they do not covary with longer maturities as stated by
Duffee [6]. Another observation is that the level is approximately constant
over the different time windows. The same relation holds for the slope and
the curvature has the same shape except for the intervals 1967 to 1977.
The next dataset is from 1982 to 2009 and the nominal yield surface
is shown in Figure 5.9 and compared to the previous set, maturities of 3
months and 6 months were included.
49
Figure 5.9: Nominal T-bond Yield Surface 1982-2009
The principal components are shown in Figure 5.10
Principal Component Analysis
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.10: T-bond 1982-2009 Principal Components
50
Here the ”hump” resurrects again and could as before be originated to
the shorter maturities. According to Table 5.7 the same pattern continues,
the level explains the most of the variation, although it has diminished
compared to the window-study above, one possible explanation for this is
the inclusion of shorter maturities.
Table 5.7: Variance Explained T-bonds 1982-2009
Component Cumulative R2
P1
78.47
P2
93.54
P3
96.60
5.1.2.2
Weekly data
For the weekly data I started out with the T-notes (1 to 10 year maturities)
from 1962-2009. The principal components are shown in Figure 5.11
Principal Component Analysis T−notes Weekly 1962 − 2009
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.11: Principal Components absolute changes Nominal Weekly Tnotes 1962-2009
51
Table 5.8: Variance Explained T-Notes Weekly 1962-2009
Component Cumulative R2
P1
91.83
P2
97.95
P3
99.42
From Table 5.8 we see that the level itself explains almost all variation.
Another observation is that explanatory power is higher than for the daily
data.
5.1.2.3
Monthly Data
The principal components of the monthly data for the period 1962-2009 is
shown in Figure 5.12
Principal Component Analysis T−notes Monthly 1962 − 2009
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.12: Principal Components absolute changes Nominal Monthly Tnotes 1962-2009
52
Table 5.9: Variance Explained T-Notes Monthly 1962-2009
Component Cumulative R2
P1
93.43
P2
98.84
P3
99.77
Table 5.9 shows that the variation explained increases when the frequency of the data decreases, e.g. from daily to monthly data.
5.1.2.4
Conclusion PCA
The results are indifferent of valuation basis, i.e. with relative or absolute changes. The explanatory power increases when the data frequency
decreases. When shorter maturities were added to the analysis, the explanatory power decreased. This can be observed by comparing the result
of LIBOR and Swaps with T-notes. It was also confirmed that the first
three Principal Components explains almost all the variation in the system.
These results are analogue with previous research.
5.1.3
Correlation with the PURCS Cycle
Since the PURCS Cycle is the driver of all different subsimulations in SAFE,
PK Air would also want it to affect the evolution of the yield curve in some
sense. Therefore the correlation between PURCS and the yields was studied.
First PURCS was compared against 1 month Nominal LIBOR, the series are
shown in Figure 5.13 and a scatter point of their absolute changes is shown
in Figure 5.14.
As can be seen in the scatter plot they do not seem to co-vary at all and
the correlation coefficient is calculated to 0.0592. So there is no correlation
between these two time series? It turned out that there actually is! The
problem lies in the scope of Signal Theory. Since the PURCS Cycle is given
on a yearly basis and the 1 month LIBOR on a daily basis, the latter will
be very noisy compared to former. Therefore the 1 month LIBOR should
be smoothed to give a fair description of the correlation. For this purpose
Smoothing Splines were used in MATLABs Curve Fitting Toolbox (cftool),
see Section 3.10. An illustration of the different smoothing levels is shown in
Figure 5.15. The next step was to analyze how the correlation was affected
by the smoothing level and the result is given in Figure 5.16. As can bee
seen the correlations is about 0.6 and to retain the most information in the
original data the smoothing factor was chosen to 0.995.
53
PURCS and 1 month Nominal Libor time series
0.12
1 month Libor
PURCS
0.1
0.08
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
1980−01−01
1990−01−01
2000−01−01
2010−01−01
Date
Figure 5.13: PURCS and 1 month Nominal LIBOR Time Series
abs PURCS vs abs 1 month Libor
0.015
0.01
0.005
0
−0.005
−0.01
−0.015
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
−3
x 10
Figure 5.14: absolute changes of PURCS vs 1 month Nominal LIBOR
54
Smoothed Libors vs Interpolated PURCS
0.12
0.1
0.08
0.06
rate
0.04
0.02
0 Smooth
0.1 Smooth
0.2 Smooth
0.5 Smooth
0.8 Smooth
0.995 Smooth
0.99964
0.999993 Smooth
1 Smooth
PURCS
0
−0.02
−0.04
−0.06
−0.08
1990−01−01
2000−01−01
Date
Figure 5.15: PURCS and 1 month Nominal LIBOR with different smoothing
Correlation with PURCS vs Smoothing Factor
0.7
0.6
0.5
ρ
0.4
0.3
0.2
0.1
0
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Smoothing Factor
Figure 5.16: Correlation between PURCS and 1 month Nominal LIBOR
with different smoothing
55
5.1.3.1
Crosscorrelation with PURCS
To determine if the correlation could be higher if the two time series were
shifted crosscorrelation was also studied for the different absolute changes of
the smoothing factors. It could be determined that the highest correlation
was obtained for lag h = 0, i.e. for the original times series not shifted. A
graphical illustration is shown in Figure 5.17.
Smooth 0 vs PURCS
Smooth 0.5 vs PURCS
0.1
1
0.05
0.5
0
0
−0.05
−0.1
−1
−0.5
0
0.5
−0.5
−1
1
lag
x 10
Smooth 0.995 vs PURCS
−0.5
0
0.5
1
4
lag
x 10
Smooth 1 vs PURCS
4
1
0.05
0.5
0
0
−0.5
−1
−0.5
0
0.5
lag
1
−0.05
−1
−0.5
0
lag
4
x 10
0.5
1
4
x 10
Figure 5.17: Crosscorrelation Coefficient between PURCS and 1 month
Nominal LIBOR with different smoothing factors
5.2
Simulation: PCA
The idea was to simulate future yield curves using the The Principal Component Representation
Xi = wi1 P1 + wi2 P2 + ... + wik Pk .
(5.2)
Since the first three principal components explained 93.68 % of the total
variation, focus was only on these and (5.2) boils down to
Xi = wi1 P1 + wi2 P2 + wi3 P3 + ,
where
∼ N (0, V ar (wi4 P4 + ... + wik Pk ))
56
(5.3)
(5.4)
To do the simulation we keep the eigenvectors, wi1 , wi2 , wi3 fixed and study
the distributions of the first three principal components, P1 , P2 , P3 .
5.2.1
5.2.1.1
LIBOR and US Swaps
Nominal
The distribution of the first PC is shown in Figure 5.18, the best fits for
the distributions are obtained with a t4.57 -location-scale1 distribution with
parameters as shown in Table 5.10. The estimation was done in MATLAB
using dfittool.
0.25
1st PC
Normal
t−location−scale
Density
0.2
0.15
0.1
0.05
0
−10
−5
0
5
10
Data
Figure 5.18: Distribution of Principal Component 1 LIBOR and Swaps
1
If the random variable X is tp distributed and µ and σ are estimated then the corresponding location-scale variable is Y = µ + σX.
57
Table 5.10: Estimate of t-location-scale PC 1 LIBOR and Swaps
Parameter Estimate Standard Error
µ
0.0089
0.029
σ
1.837
0.032
ν
4.571
0.300
The distribution of the second PC is shown in Figure 5.19 and the parameter estimation in Table 5.11
0.45
PC 2
Normal
t−location−scale
0.4
0.35
Density
0.3
0.25
0.2
0.15
0.1
0.05
0
−15
−10
−5
0
5
10
15
Data
Figure 5.19: Distribution of Principal Component 2 LIBOR and Swaps
58
Table 5.11: Estimate of t-location-scale PC 2 LIBOR and Swaps
Parameter Estimate Standard Error
µ
0.0680
0.014
σ
0.7841
0.016
ν
2.014
0.073
For the third component the following t-location-scale estimation was
obtained
Table 5.12: Estimate of t-location-scale PC 3 LIBOR and Swaps
Parameter Estimate Standard Error
µ
-0.003
0.0059
σ
0.33
0.0074
ν
1.89284
0.071
From these findings the simulation was done by Monte Carlo simulation
from the corresponding t-location scale distributions. The number of days
simulated was 8000 days (about 16 years) since this is the average length
of PK Air’s deals. To make the simulations more stable the simulations are
done 1000 times and then averaged. The positive thing with averaging is
that the simulations become more stable, but the negative thing is that the
yields flatten out and lose some of their original shape.
The simulation is started from the following yield curve from the simulaTable 5.13: Fictional Starting Yield Curve
TTM
yield
1m
0.01
2m
0.015
3m
0.0175
6m
0.02
1yr
0.023
2yr
0.025
3yr
0.027
4yr
0.028
5yr
0.029
7yr
0.0295
10yr
0.0296
∗
tion according to (5.3) the normalized changes Xsim
were obtained. These
were tranformed back to Xsim using
∗
Xsim = Xsim
· V (Xi ) + E (Xi )
(5.5)
where Xi is the original changes column i, e.g. the column for changes of
3 month LIBOR over time. The next step was to transform back the yield
data by
Si+1 = ∆i,i+1 + Si
(5.6)
where Si are the yields at time i and ∆i,j is the absolute yield change from
time i to j. A simulation for 8000 days conducted 1000 times and then
averaged resulted in the following yield surface
The result is not that pleasing since the yields go negative. The surface
59
Figure 5.20: Simulated Surface for 8000 days Nominal LIBOR and Swaps
also shows a hump for shorter maturities. Lots of the movements have been
flattened out because of the averaging. A Principal Component Analysis on
the simulated changes correlation matrix is visible in Figure 5.21.
Principal Component Analysis Simulated Libor and Swaps
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.21: Principal Components Simulated Nominal LIBOR and Swaps
60
If comparing with Figure 5.5 the level component have shifted from approximately -0.4 to 0.4, the slope has also been inverted, but the curvature
remains in the same setting. The correlation matrix of the simulated data
is shown in Table 5.14
Table 5.14: Correlation Matrix Simulated Nominal LIBOR and Swaps
1 m
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1 m
1
0.975
0.662
0.324
0.104
-0.015
-0.036
-0.030
-0.047
-0.042
-0.034
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1
0.811
0.525
0.320
-0.023
-0.057
-0.060
-0.079
-0.078
-0.074
1
0.923
0.813
0.025
-0.035
-0.063
-0.081
-0.090
-0.098
1
0.974
0.064
-0.002
-0.041
-0.055
-0.069
-0.082
1
0.113
0.049
0.006
-0.004
-0.020
-0.036
1
0.998
0.994
0.993
0.991
0.989
1
0.999
0.998
0.998
0.996
1
0.999
0.999
0.998
1
0.999
0.999
1
0.999
1
61
5.2.1.2
Real
The normalized original change matrix X ∗ had the setting shown in Table
5.15 The simulation was conducted in the same manner as earlier and it
Table 5.15: Correlation Matrix Real LIBOR and Swaps
1 m
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1 m
1
0.876
0.869
0.839
0.794
0.677
0.667
0.662
0.662
0.666
0.673
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1
0.901
0.873
0.822
0.691
0.677
0.670
0.674
0.678
0.683
1
0.938
0.893
0.744
0.728
0.720
0.725
0.728
0.732
1
0.928
0.746
0.733
0.723
0.727
0.729
0.732
1
0.738
0.725
0.714
0.717
0.718
0.720
1
0.974
0.959
0.955
0.944
0.920
1
0.974
0.971
0.962
0.938
1
0.974
0.967
0.944
1
0.983
0.967
1
0.975
1
yielded the yield curve in Figure 5.22. It showed the same pattern as for the
nominal yields, although not as neagtive, but still inplausible.
Figure 5.22: Simulated Surface for 8000 days 1000 times Real LIBOR and
Swaps
A PCA on the simulated changes yielded the following results
The variance explained is displayed in Table 5.16 The simulated correlation
matrix is given in Table 5.17
62
Principal Component Analysis Simulated Libor and Swaps
0.6
1st PC
2nd PC
3rd PC
0.4
Change in yield
0.2
0
−0.2
−0.4
−0.6
−0.8
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.23: Principal Components Simulated Real LIBOR and Swaps
Table 5.16: Variance Explained Simulated Real LIBOR and Swaps 2008 2009
Component Cumulative R2
P1
89.17
P2
94.88
P3
100
5.2.2
US Treasuries Constant Time to Maturity
For comparison purposes the US Treasuries were studied. First the longest
set of data is analyzed, that is from 1962 to 2009. For this interval T-notes
were only available with maturities of 1, 3, 5 and 10 years. The correlation
matrix had the following setting The simulated surface can be seen in Figure
5.24
The resulting surface for this setting is more pleasing than for the LIBOR
and Swaps rates. An upward trend is apparent. PCA of the simulated data
are shown in Figure 5.25 and the correlation matrix in Table 5.19
From these results the conclusion was that the simulations were highly
dependent on the propagation of the historical datas used as a foundation
for the simulations. For the LIBOR and Swaps there has been a downward
trend since 1988 and the simulations therefore continue in the same pattern.
For the T-notes the historical data were Λ-shaped but with more upward
trends than downward and the simulations exhibit an upward trend.
63
Surface for 8000days
0.06
0.05
rate
0.04
0.03
0.02
0.01
0
10
5
0
Term to maturity (years)
6000
4000
2000
0
8000
Days
Figure 5.24: Simulated Surface for Nominal T-notes 1962-2009
Principal Component Analysis Simulated T−notes 62−09
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.25: Principal Components Simulated Yields
64
Table 5.17: Correlation Matrix Simulated Real LIBOR and Swaps 1988 2009
1 m
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1 m
1
0.982
0.86
0.730
0.595
0.768
0.765
0.770
0.770
0.776
0.789
2 m
3 m
6 m
1 yr
2 yr
3 yr
4 yr
5 yr
7 yr
10 yr
1
0.938
0.846
0.736
0.832
0.825
0.826
0.825
0.830
0.842
1
0.978
0.925
0.892
0.879
0.872
0.872
0.874
0.882
1
0.984
0.868
0.852
0.842
0.842
0.842
0.846
1
0.822
0.803
0.789
0.790
0.788
0.790
1
0.999
0.998
0.998
0.998
0.998
1
0.999
0.999
0.999
0.999
1
0.999
0.999
0.999
1
0.999
0.999
1
0.999
1
Table 5.18: Correlation Matrix Nominal T-notes 1988 - 2009
1 y
3 y
5 y
10 y
5.3
1 y
1
0.847
0.800
0.729
3 y
5 y
10 y
1
0.941
0.873
1
0.926
1
Simulation: Rebonato
Since the simulation from the Principal Component Representation was
mostly unplausible a test with the simulation conducted by Rebonato, Mahal, Joshi, Buchholz and Nyholm [17] was also implemented. I started out
with their first approach; the naive. As described earlier it is an historical
simulation method, where the future yields are simulated from the historical
yield changes. The method is as follows:
1. Draw a random starting yield change curve from the historical yield
curve changes.
2. Draw a random number between 5 and 50 for the window length of
the subsimulation.
3. In each extraction of a yield curve change there is a 5 % probability
of jumping out of the window and go back to step 1.
4. Continue with the subsimulations until the desired simulation length
is obtained.
5.3.1
T-notes
US Treasury yields with maturities of 3 months to 10 years between 2nd
of January 1986 to 7th of May 2004 were studied to be able to relate to
the study by Rebonato and Nyholm [18]. The difference from the other approach was that no averaging was done and percentage changes were used to
keep the yields positive. One yield curve outcome is shown in Figure 5.26.
The Principal Components before and after are shown in Figures 5.27 and
65
Table 5.19: Correlation Matrix Simulated Nominal T-notes 1988 - 2009
1y
3y
5y
10 y
1y
1
3 y 0.586
1
5 y 0.417 0.916
1
10 y 0.215 0.581 0.852
1
Figure 5.26: Simulated Nominal Surface T-yields 1962 - 2009
5.28. As can be seen the components are preserved. To clarify this the
correlation matrices are shown in Table 5.20 and 5.21. As stated in Rebonato, Mahal, Joshi, Buchholz and Nyholm [17] this procedure asymptotically
recovers the correlation matrix and the same holds for the eigenvalues and
eigenvectors.
The final result with this naive approach is not pleasing although that the
correlation structure and the principal components are recovered. Therefore
the author’s spring and mean reversion model was tried. It uses spring constants and mean-reversion levels to match the variance of the curvature of
the simulation with the empirical variance of the curvature. It is not clear
66
Principal Component Analysis Treasury yields 1986−2004
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.27: Principal Components Treasury Yields 1982 - 2004
Principal Component Analysis Simulated yields
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure 5.28: Principal Components Simulated Yields
67
Table 5.20: Correlation Matrix Nominal Treasury Yields 1982 - 2004
3 m
6 m
1 y
2 y
3 y
5 y
7 y
10 y
3 m
1
0.786
0.053
0.038
0.037
0.038
0.043
0.045
6 m
1 y
2 y
3 y
5 y
7 y
10 y
1
0.057
0.047
0.042
0.039
0.043
0.042
1
0.889
0.860
0.816
0.769
0.730
1
0.964
0.908
0.859
0.810
1
0.953
0.912
0.871
1
0.964
0.940
1
0.974
1
Table 5.21: Correlation Matrix Simulated Nominal Treasury Yields
3 m
6 m
1 y
2 y
3 y
5 y
7 y
10 y
3 m
1
0.790
0.054
0.041
0.040
0.043
0.049
0.051
6 m
1 y
2 y
3 y
5 y
7 y
10 y
1
0.070
0.057
0.047
0.044
0.050
0.047
1
0.888
0.858
0.811
0.766
0.721
1
0.962
0.899
0.853
0.802
1
0.951
0.909
0.866
1
0.961
0.936
1
0.973
1
from their paper on how to do this matching. In this study a non-linear
least squares routine was tried on the data. Unfortunately, the results were
still implausible, the final curvatures obtained translated distributions, either greater or less than zero. This yielded even more implausible results
than the naive approach. Therefore this model was abandoned.
5.4
Simulation: Brute Force Model
Since the two former models did not result in plausible future yield curves,
a need for a new model arose. Another criteria was that the yield curves
should be affected by the PURCS cycle. As stated in the article by Rebonato [17] - Brute-force simulation and analysis of the results is therefore
often the only possible investigative route. Due to the failure of the former
models, the focused shifted to brute force modeling. The idea was to add
the cycle as a factor to the yield curve evolution. To keep the correlation
structure smoothing splines were applied to the LIBOR and Swaps yields
with a smoothing factor of 0.995, see Section 5.1.3. The relationship between the PURCS cycle and the smoothed yields was set up in the following
way
a (j) + log (y (i, j)) = log (1 + c (i) · d (j))
(5.7)
where a (j) is a translation factor, y (i, j) is the yield at date i and maturity
j, c (i) is the PURCS cycle at date i and d (j) is a scaling factor. 1 is added
to the cycle since it has negative values. The reason that there is no scaling
factor on the yields is that this is covered by the scaling on the PURCS
cycle. If a scaling factor is introduced the optimization routine fails to find
plausible solutions, it will just keep on rescaling.
This relationship was studied for historical data for the LIBOR and Swaps
68
and the PURCS cycle. Expression (5.7) was re-expressed as
F (i, j) = a (j) + log (y (i, j)) − log (1 + c (i) · d (j))
(5.8)
To determine the functions a and d, for each maturity j a non-linear least
squares optimization routine, lsqnonlin, in MATLAB was used. The results
are shown in Figure 5.29 and 5.30
Optimized a
3.5
3.4
3.3
a
3.2
3.1
3
2.9
2.8
0
1
2
3
4
5
6
7
8
9
10
Term to Maturity (years)
Figure 5.29: Least Squares optimized a
As can be seen, a do not vary that much and is therefore set to 3. For d an
exponential function
(5.9)
d (tj ) = m + pe−rtj
was fitted with least squares to the function to describe the factor variation
across maturities. The fitted function is shown in Figure 5.31 and its values
in Table 5.22 and Goodness of Fit values in Table 5.23.
Table 5.22: Estimated
bounds
parameter
m
p
r
parameters for function d with 95 % Confidence
value
0.4152
5.835
0.2812
Lower Bound
-0.211
5.245
0.2067
69
Upper Bound
1.041
6.426
0.3557
Optimized d
6
5
d
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Term to Maturity (years)
Figure 5.30: Least Squares optimized d
d with fitted function
d
fit
6
5.5
5
d(TTM)
4.5
4
3.5
3
2.5
2
1.5
1
0
1
2
3
4
5
6
7
8
9
10
Term to Maturity (years)
Figure 5.31: Least Squares fitted function to d
70
Table 5.23: Goodness of Fit for function d
Measure
Value
Sum of Squared Errors 0.3051
R-Squared
0.9926
Adjusted R-Squared
0.9907
Having determined a and d, equation (5.8) could be rewritten as
F (i, j) − a = log (y (i, j)) − log (1 + c (i) · d)
eF (i,j)−a = elog(y(i,j))−log(1+c(i)·d)
y (i, j)
=
(1 + c (i) · d)
(5.10)
(5.11)
(5.12)
in other words, eF (i,j)−a is the yields without PURCS cycle influence, and
that is what should be simulated. From this it is clear that we only need
the parameter d, since the value of a does not affect the expression.
The last equation could now be re-expressed as
z (i, j) =
y (i, j)
(1 + c (i) · d)
(5.13)
where z is referred to as the Cycle Neutral Yield at date i and maturity
j. By using this ”logarithmic approach” the PURCS cycle is affecting the
yields as a factor. The Cycle Neutral surface is shown in Figure 5.32.
The next step was to apply Nelson-Siegel’s curve fitting technique to this
surface. This was conducted by running an nonlinear least squares routine,
lsqnonlin, in MATLAB. It was done with the same conditions as stated in
Section 3.17. The time series of the four parameters are shown in Figure
5.33, and the error of the fit in Figure 5.34. The downward trend since 1988
to 2009 is evident in β0 .
71
Figure 5.32: Historical PURCS Cycle Neutral Yield Surface
Time Series β0
Time Series β1
0.2
0
0.15
−0.02
0.1
−0.04
0.05
0
0
2000
4000
6000
8000
−0.06
0
Time Series β2
2000
4000
6000
8000
Time Series τ
0.05
2.002
2.001
0
2
1.999
−0.05
0
2000
4000
6000
8000
1.998
0
2000
4000
6000
8000
Figure 5.33: Nelson-Siegel estimated parameters from z
72
Nelson−Siegel SSE Error
−4
1.2
x 10
1
SSE
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
7000
8000
Figure 5.34: Nelson-Siegel Sum of Squares fit error
How to model these series? Since we are still in a world of low yields
post the financial crisis, it is fairly reasonable to expect that the yields will
rise with time. PK Air had also expressed a desire to have mean-reversion
in the model and therefore Ornstein-Uhlenbeck processes are applied, see
Section 3.16.
By looking at the time series in Figure 5.33, it seemed as τ was a combination of β0 and β1 . If this is possible simulation time can be reduced.
Therefore the following relationship was tried on historical data with a multiple linear regression. The obtained parameters are shown in Table 5.24
and the Goodness of Fit values in Table 5.25.
τ = v + qβ0 + sβ1
(5.14)
The following parameter estimates were obtained
Table 5.24: Estimated
Confidence bounds
parameter
v
q
s
parameters for fitting β0 and β1 to τ with 95 %
value
2.0021
-0.0365
-0.0367
Lower Bound
2.0021
-0.0370
-0.0374
73
Upper Bound
2.0022
-0.0360
-0.0360
Table 5.25: Goodness of Fit for regression
Measure
Value
Sum of Squared Errors 8.7159e-004
R-Squared
0.7712
Adjusted R-Squared
0.7711
5.4.1
Calibration of Ornstein-Uhlenbeck processes
To Calibrate the processes Maximum-Likelihood Estimation was applied,
see Section 3.13 for a general description and Appendix B for a derivation of
the estimates for the Ornstein-Uhlenbeck process. Since PK Air wants the
simulation to be as fast as possible, the data set of estimated Nelson-Siegel
parameters were chosen to be represented by 30 points instead, with equal
distance in between themselves. Later when the simulation is conducted
more frequent data sets can be obtained by linear interpolation. The Maximum Likelihood parameter estimation results are shown in Tables 5.26, 5.27,
and 5.28.
Table 5.26: Estimated Ornstein-Uhlenbeck parameters β0
parameter value
µ
0.0526
λ
0.0994
σ
0.0053
Table 5.27: Estimated Ornstein-Uhlenbeck parameters β1
parameter
value
µ
-0.0279
λ
0.7090
σ
0.0133
The condition in Nelson Siegel’s model that β0 + β1 > 0 had still to
be governed. To cope with these negativity problems, a help function was
introduced as soon as their sum was smaller than 0.01. If this study has been
done 2 years ago, it would probably have been unnecessary with this ”pillow
function”, but since the financial crisis and its aftermath is considered as an
extreme event, these kind of helping functions is justified. If the sum the
74
Table 5.28: Estimated Ornstein-Uhlenbeck parameters β2
parameter
value
µ
-0.0077
λ
1.7382
σ
0.0274
two parameters is defined as
f (i) = β0 (i) + β1 (i)
(5.15)
Then the help function g (i) is
(
f (i)
ae a −1 ,
if f (i) < 0.01
g(i) =
f (i) ,
if f (i) ≥ 0.01
a graphical illustration of the function is given in Figure 5.35. As can bee
seen it is linear above the threshold of 0.01.
The ”help” from the function is then portioned out in the following way
Help function to avoid negative yields
0.06
0.05
g(f(i))
0.04
0.03
0.02
0.01
0
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
f(i)
Figure 5.35: Help function with threshold 0.01
β0 (i)
(β0 (i) + abs (β1 (i)))
β1 (i)
β1 (i) =
(β0 (i) + abs (β1 (i)))
β0 (i) =
f (i)
· ae a −1 − f (i) + β0 (i)
(5.16)
f (i)
· ae a −1 − f (i) + β1 (i)
(5.17)
as can bee seen from the equations above the help is related to their respective proportions. After non-negativity is assured, the next step is to
75
generate the yields with the Nelson-Siegel function, see equation 3.44, the
results are shown in Figures 5.36 and 5.37.
Figure 5.36: Simulated PURCS Cycle Neutral Yield Surface
The final step of the simulation is to add a simulated future PURCS Cycle
influence, by rearranging equation (5.13) to
y (i, j) = z (i, j) · (1 + c (i) · d)
the final result is shown in Figures 5.38 and 5.39
76
(5.18)
Simulated Nelson−Siegel Time Series
0.06
1m
2m
3m
6m
1y
2y
3y
4y
5y
7y
10y
0.05
yield
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
5000
6000
Simulated Days
Figure 5.37: Simulated PURCS Cycle Neutral Yield Time Series
Figure 5.38: Final Simulated Yield Surface
77
Final Simulated Time Series
0.07
1m
2m
3m
6m
1y
2y
3y
4y
5y
7y
10y
0.06
0.05
yield
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
5000
6000
Simulated Days
Figure 5.39: Final Simulated Yield Time Series
To produce statistics over the simulations, 10000 yield surfaces were generated and percentiles were calculated to visualize the results. The results
for the 1 month, 1 year and 10 year term to maturities over the 30 time
steps are shown in Figures 5.40, 5.41 and 5.42. The Cyclical behavior is as
expected more evident in the shorter maturities.
The same results are available for all term to maturities and can in turn be
used for PK Air’s Swap Breakage Scenario calculation.
78
Percentiles 1 month yields
0.08
2.5
25
50
75
97.5
0.07
0.06
yield
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20
25
30
Figure 5.40: Percentiles for 10 000 simulations of 1 month yield
Percentiles 1 year yields
0.08
2.5
25
50
75
97.5
0.07
0.06
yield
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20
25
30
Figure 5.41: Percentiles for 10 000 simulations of 1 year yield
79
Percentiles 10 year yields
0.08
2.5
25
50
75
97.5
0.07
yield
0.06
0.05
0.04
0.03
0.02
0
5
10
15
20
25
30
Figure 5.42: Percentiles for 10 000 simulations of 10 year yield
80
Chapter 6
Discussion
6.1
Conclusions
With Principal Component Analysis, the results were analogue with precedent studies. The first three components are enough to describe the variation
of the yield curve and the inclusion of shorter maturities decreases the explanatory power.
Long horizon simulations of the yield curve from the Principal Component
Representation did not yield plausible results. The term structure loses its
shape and this historical simulation procedure exhibits the same trend as
the historical data. This trend results in either negative yields or a ”zero
carpet” depending on the valuation basis.
The ”Rebonato approach” also did not yield plausible results. The problem with this approach was the implementation of the spring constants by
matching the simulated variance of the curvature to the historical ones. It
is not stated in their paper on how to do this, and efforts in this thesis failed.
Because of the lack of long-term models in the Real World Measure the
Brute Force Model was developed. The model manages to produce plausible future yield curve scenarios with cyclical influence, which was demanded
by PK Air since it influences all other parts of their model. The model also
follows the other criteria specified by PK Air: tractability, few parameters
and fast to simulate. For instance, it is easy to edit the input parameters
of the Ornstein-Uhlenbeck processes with PK Air’s view of the future. The
help function to assure non-negativity is justified since the financial crisis
is an extreme event - and must be treated thereafter. This author believes
that this function is better than ordinary methods such as the absorption
and reflection methods.
81
The model could be even better with some refinement and development.
Maybe some people argue that the PURCS Cycle effect is to large and that
it could be wise to investigate another way of linking the yields to the cycle
than what has been done here. Some information is also lost by smoothing
the yields, representing frequent data with fewer points, fitting functions
and manipulating the data back and forth. This could all be added to a
”noise part” in the simulation that could be added in the end, but this
needs further investigation.
The Ornstein-Uhlenbeck process could probably be replaced by an even more
sophisticated process, e.g. with varying volatility, mean-reversion level and
speed of mean-reversion.
The fitting techniques with least squares, regression and maximum-likelihood
could also be inspected and developed. The same holds for the Nelson-Siegel
model, there is an extension of the model - the Nelson-Siegel and Svensson
model, which introduces another parameter and allows for a second ”hump”
on the curve. But as always one is limited by time so this lies outside of the
scope of this thesis.
Correlation is often criticized of being a limited measure of dependence.
It could be interesting to look at other dependence measures, e.g. cointegration and copulas.
The data set used for the Brute Force Model is the US LIBOR and Swaps
rates, it would be interesting to estimate the model from a longer set of
data, e.g. the Treasury Yields, but they have the drawback of not having
the shorter maturities for the longer period. Another topic for discussion is
how valid the data from 1962 are? Has the financial markets changed since
then in terms of globalization and size? This author believes that the US
LIBOR and Swaps are more relevant, although the longer period Treasury
Yields could serve as a reference.
6.2
Future Studies
During the work with this thesis many ideas and questions arose, some of
them have been stated in the previous section. Unfortunately these lie out
of the scope of this thesis. Some of these are:
• Investigate if the use of the Nelson-Siegel and Svensson model improves
the model and also consider other curve fitting techniques.
• Study the estimation techniques used and see if these could be replaced or refined by even better ones, e.g. Maximum Likelihood, Least
82
Squares and Regression.
• Evaluate the link between the yield curve and the business cycle - are
there other methods?
• Develop or replace the Ornstein-Uhlenbeck processes.
• Evaluate the forecasting ability of the model.
• Estimate the model from other data sets, e.g. other lengths and countries.
• Investigate other dependence measures than correlation, for instance
cointegration and copulas.
To conclude this model presents one approach for simulating yield curves
under the influence of a business cycle. From the research by this author
this is one of the first in this field. Therefore this author hopes it could serve
as a foundation for further developments and refinements.
83
84
Appendix A
PCA Results
Here are the results of PCA on T-notes between 1962 and 2009 with 5 year
windows.
Principal Component Analysis T−notes 1962 − 1967
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.1: Principal Components T-notes 1962-1967
85
10
Principal Component Analysis T−notes 1967 − 1972
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
1
2
3
4
5
6
7
8
9
10
Time to maturity
Figure A.2: Principal Components T-notes 1967-1972
Principal Component Analysis T−notes 1972 − 1977
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.3: Principal Components T-notes 1972-1977
86
10
Principal Component Analysis T−notes 1977 − 1982
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.4: Principal Components T-notes 1977-1982
87
10
Principal Component Analysis T−notes 1982 − 1987
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.5: Principal Components T-notes 1982-1987
88
10
Principal Component Analysis T−notes 1987 − 1992
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.6: Principal Components T-notes 1987-1992
89
10
Principal Component Analysis T−notes 1992 − 1997
0.8
1st PC
2nd PC
3rd PC
0.6
Change in yield
0.4
0.2
0
−0.2
−0.4
−0.6
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.7: Principal Components T-notes 1992-1997
90
10
Principal Component Analysis T−notes 1997 − 2002
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.8: Principal Components T-notes 1997-2002
91
10
Principal Component Analysis T−notes 2002 − 2007
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.9: Principal Components T-notes 2002-2007
92
10
Principal Component Analysis T−Notes 2007−2009
1
1st PC
2nd PC
3rd PC
0.8
Change in yield
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1
2
3
4
5
6
7
8
9
Time to maturity
Figure A.10: Principal Components T-notes 2007-2009
93
10
94
Appendix B
Maximum Likelihood
Estimation
Ornstein-Uhlenbeck
From the solution from Section 3.16
Xt = θ + e
−κ(t−s)
Z
t
(Xs − θ) + σ
e−κ(t−u) dWu
s
the conditional mean and variance can be derived
E [Xt | Xs ] = θ + e−κδ (Xs − θ)
where δ has been introduced for the time step (t − s).
Z t
−κ(t−u)
e
dWu
V ar [Xt | Xs ] = E σ
s
= (by Itô Isometry)
Z t
2
−2κ(t−u)
=E σ
e
du
s
σ2 =
1 − e−2κδ
2κ
Therefore Xt is normally distributed with E [Xt | Xs ] = θ + e−κδ (Xs − θ)
2
and V ar [Xt | Xs ] = σ2κ 1 − e−2κδ .
From this the conditional probability density function for Xi+1 given Xi
with time step δ as
"
2 #
Xi − Xi−1 e−κδ − θ 1 − e−κδ
1
f (Xi+1 | Xi ; κ, θ, σ) = √
exp −
2σ̂ 2
2πσ̂ 2
95
where
σ̂ 2 = σ 2
1 − e−2κδ
2κ
The log-likelihood function for a sample X0 , X1 , ..., Xn is derived from the
conditional density function [22]
` (κ, θ, σ̂) =
n
X
logf (Xi | Xi−1 ; κ, θ, σ̂)
i=1
n
i2
1 Xh
n
Xi − Xi−1 e−κδ − θ 1 − e−κδ
= − log (2π) − n · log (σ̂) − 2
2
2σ̂
i=1
To find the maximum of the log-likelihood surface the partial derivatives is
set to zero and solved for
n
i
d` (κ, θ, σ̂)
1 Xh
Xi − Xi−1 e−κδ − θ 1 − e−κδ
=0 = 2
dθ
2σ̂
i=1
Pn −κδ
i=1 Xi − Xi−1 e
⇒θ=
n (1 − e−κδ )
n
i
d` (κ, θ, σ̂)
δe−κδ X h
2
−κδ
=0 = −
(X
−
θ)
(X
−
θ)
−
e
(X
−
θ)
i
i−1
i−1
dκ
2σ̂ 2
i=1
Pn
(Xi − θ) (Xi−1 − θ)
1
⇒ κ = − log i=1
Pn
2
δ
i=1 (Xi−1 − θ)
n
i2
d` (κ, θ, σ̂)
n
1 Xh
=0 = − 3
Xi − Xi−1 e−κδ − θ 1 − e−κδ
dσ̂
σ̂ σ̂
i=1
n
i2
1 Xh
2
⇒ σ̂ =
Xi − Xi−1 e−κδ − θ 1 − e−κδ
n
i=1
The conditions depend on each other. Although θ and κ are independent of
σ̂. Therefore θ can be found be substituting κ into the expression for θ.
96
To faciliate some notations are introduced
Sx =
n
X
Si−1
i=1
Sy =
Sxx =
n
X
i=1
n
X
Si
2
Si−1
i=1
Sxy =
Syy =
n
X
i=1
n
X
Si−1 Si
Si2
i=1
results in
Sy − e−κδ SX
n (1 − e−κδ )
θ=
1 Sxy − θSx − θSy + nθ2
κ = − ln
δ
Sxx − 2θSx + nθ2
substituting κ into θ yields
Sy −
nθ =
1−
Sxy −θSx −θSy +nθ2
Sxx −2θSx +nθ2
Sxy −θSx −θSy +nθ
Sxx −2θSx +nθ2
SX
2
remove denominators
Sy Sxx − 2θSx + nθ2 − Sxy − θSx − θSy + nθ2 Sx
nθ =
(Sxx − 2θSx + nθ2 ) − (Sxy − θSx − θSy + nθ2 )
collecting terms
(Sy Sxx − Sx Sxy ) + θ Sx2 − Sx Sy + θ2 n (Sy − Sx )
nθ =
(Sxx − Sxy ) + θ (Sy − Sx )
and this yields
nθ (Sxx − Sxy ) − θ Sx2 − Sx Sy = (Sy Sxx − Sx Sxy )
and the final expressions are obtained as
θ=
(Sy Sxx − Sx Sxy )
n (Sxx − Sxy ) − (Sx2 − Sx Sy )
97
1 Sxy − θSx − θSy + nθ2
κ = − ln
δ
Sxx − 2θSx + nθ2
σ̂ 2 =
i
1h
Syy − 2αSxy + α2 Sxx − 2θ (1 − α) (Sy − αSx ) + nθ2 (1 − α)2
n
with
σ 2 = σ̂ 2
2λ
1 − α2
and
α = e−κδ
98
Bibliography
[1] C. Alexander. Market Models - A Guide to Financial Data Analysis.
John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West
Sussex PO19 8SQ, England, 2001.
[2] British Bankers Association. Libor rate. http://www.bbalibor.com/
bba/jsp/polopoly.jsp?d=1625.
[3] C. Bernadell, J. Coche, and K. Nyholm. Yield curve prediction for
the strategic investor. Technical report, Risk Management, European
Central Bank, 2005.
[4] G. Blom, J. Enger, G. Englund, J. Grandell, and L. Holst. Sannolikhetsteori och statistikteori med tillämpningar. Studentlitteratur, Studentlitteratur, Lund, 2005.
[5] D. Bolder and D. Stréliski. Yield curve modelling at the bank of canada.
Technical report, Bank of Canada, 1999.
[6] G. R. Duffee. Idiosyncratic variation of treasury bill yields. The Journal
of Finance, 2:527–550, 1996.
[7] R. Fiori and S. Iannotti. Scenario based principal component valueat-risk: An application ot italian banks’ interest rate risk exposure.
Technical report, Banca d’Italia, 2006.
[8] Financial Trend Forecaster.
Us historical inflation
http://inflationdata.com/Inflation/Inflation_Rate/
HistoricalInflation.aspx.
data.
[9] O. Hinton. Describing random sequences. http://www.staff.ncl.ac.
uk/oliver.hinton/eee305/Chapter6.pdf.
[10] J. Hull. Options, Futures and other Derivative Securities, 2nd Ed. Prentice Hall, Englewood Cliffs, New Jersey 07632, 1993.
[11] R. Litterman and J. Scheinkman. Common factors affecting bond returns. The Journal of Fixed Income, pages 54–61, June 1991.
99
[12] D. G. Luenberger. Investment Science. Oxford University Press, Inc.,
198 Madison Avenue, New York, NY 10016, USA, 1998.
[13] Mathworks. Curve fitting toolbox user’s guide. Technical report, Mathworks, 2008.
[14] C. R. Nelson and A.F. Siegel. Parsimonious modeling of the yield
curves. Journal of Business, 60:473–489, 1987.
[15] Federal Reserve Bank of St. Louis. Economic research.
research.stlouisfed.org/fred2/categories/115.
http://
[16] W. Phoa. Yield curve risk factors: Domestic and global contexts. Technical report, The Capital Group of Companies.
[17] R. Rebonato, S. Mahal, M. Joshi, L-D. Buchholz, and K. Nyholm.
Evolving yield curves in the real-world measures: a semi-parametric
approach. The Journal of Risk, 7:29–61, 2005.
[18] R. Rebonato and K. Nyholm. Long horizon yield curve forecasts: Comparison off semi-parametric and parametric approaches. Applied Financial Economics, 18:1597–1611, 2008.
[19] A. Santomero and D. Babbel. Financial Markets, Instruments and
Institutions, Irwin McGraw-Hill. Irwin McGraw-Hill, Boston, Massachusetts, 1997.
[20] K. P. Scherer and M. Avellaneda. All for one... one for all?: A principal
component analysis of latin american brady bond debt from 1994 to
2000. Technical report, Courant Institute of Mathematical Sciences,
New York University, United States, 2000.
[21] G. M. Soto. Using principal component analysis to explain term structure movements: Performance and stability. Progress in Economics
Research, 8:203–225, 2004.
[22] M.A. van den Berg. Ornstein-uhlenbeck process. http://www.sitmo.
com/doc/Ornstein-Uhlenbeck_process.
100